一类抛物型分布参数系统的边界控制

利用经典李对称理论,研究一类抛物型分布参数系统的边界控制问题,分别设计开环和闭环形式的边界控制律,实现系统状态的定态控制。借助于无穷小生成元作为分析工具,应用微分方程的不变性条件,确定系统经典李对称的具体表示形式,即其所对应的无穷小生成元表达式。之后,分别针对开环和闭环控制结构,设计出系统解析形式的边界控制条件。通过设定系统参数、初始条件和控制目标,开环和闭环边界控制都能实现设定的控制要求。相比较而言,开环控制的输出误差收敛速度较慢; 闭环控制收敛速度较快,不过入口附近有无法完全避免的超调现象。提供的研究结果,对于一类包含传导和对流特性的温度或浓度模型的定态控制问题有一定指导意义。     

一类热方程的边界控制及不确定性分析

基于对称群理论中的微分方程对称形式,研究一类热方程的边界控制问题,以及当边界条件中包含不确定控制系数和未知边界扰动时,系统控制目标的变化问题。首先,利用微分方程对称分析了求解热方程无穷小生产元的过程,进而利用所得的无穷小生产元和不变性条件,通过设定系统初始条件、边界条件和控制目标,设计了未含不确定性时热方程的边界控制条件,实现系统状态稳定于设定常值。相应结论进行了系统仿真,仿真结果验证了设计条件的正确性。然后详细分析了加入不确定控制系数和边界扰动后,系统状态的变化情况,并给出分析结论。设定控制系数,对于不同形式的边界扰动、初始条件和控制目标,进行了系统仿真,仿真结果验证了所得的分析结论。     

基于连续体模型的人群疏散系统的边界控制

研究一类存在扰动的一维人群疏散系统的边界控制问题.以走廊中的人群动态为例,基于数量守恒定律建立人群动态模型;由非线性偏微分方程描述系统模型,并直接在分布参数的范畴内,设计Robin、Neumann、Dirichlet三种边界控制律,用于控制行人在疏散过程中的移动方式,避免拥堵的产生;利用李雅普诺夫方法对边界控制律作用下的人群疏散系统稳定性给出详细证明,并通过一个仿真实例验证边界控制律的有效性.研究成果可以应用到生活中单入口单出口场所的人群动态管理.     

状态相关闭排队网络中的性能指标灵敏度公式

本文通过研究一类Ｍａｒｋｏｖ过程在无穷小矩阵的参数摄动下稳态性能指标的灵敏度,运用无穷小矩阵的群逆,实现矩阵和势能这三个描述稳定性能指标灵敏度的等价量,给出了状态相关闭排队网络在参数摄动下的稳态性能指标灵敏度公式,这些结果可直接用于排队网络的控制和优化。     

Fokker-Planck方程的非古典势对称群及新显式解

本文利用一种新方法对Fokker- Planck方程的非古典势对称群生成元进行研究,找到方程的几个非古典势对称群生成元,并采用非古典对称群方法由这些对称群生成元构造得到Fokker- Planck方程的相应显式解.这些新显式解不能由Fokker -Planck方程本身的Lie对称或Li-e B cklund对称来获得.在验证所求得显式解的过程中,还发现并得到了另外几个显式解.这些新显式解则不能由Fokker -Planck方程本身的Lie对称,Lie- B cklund对称或非古典势对称来获得.文章表明,通过偏微分方程的非古典势对称群生成元来寻找其显式解是可能的.     

基于正交函数的立式淬火炉控制系统参数辨识

大型立式淬火炉体积庞大,工况复杂,炉内温度分布呈本征非均匀性.为了获得温度控制高精度和高均匀性提出参数辨识算法,包括求解正交函数正、反向积分运算矩阵,以块脉冲函数为基函数利用正交函数变换将由偏微分方程描述的分布参数系统模型转化为最小二乘形式的代数方程.辨识过程中考虑了大型立式淬火炉温度分布参数系统模型边界条件和初始条件的影响,提高了参数辨识精度,算法计算量小且保持了系统的空间分布特性.     

末端有未知扰动的分布参数柔性机械臂的鲁棒边界控

本文研究在柔性机械臂的末端具有未知扰动的边界控制,以降低机械臂的振动.柔性机械臂的动态特性由偏微分方程表示的分布参数模型描述.在机械臂的末端边界基于Lyapunov直接法进行控制,以调节机械臂的振动.应用本文所提出的边界控制方法,可达到外界干扰下的指数稳定性.所提出的控制方法与系统参数无关,可确保在参数变化下系统具有鲁棒性.最后对所提控制方法的有效性进行了数值模拟.     

双曲线型分布参数系统边界控制下的干扰解耦

针对干扰输入对系统输出的影响, 研究双曲线型分布参数系统在边界控制下的干扰解耦问题. 首先应用有界控制算子等价转换方法, 获取边界控制下双曲线型分布参数系统的有界拓展形式; 然后通过伽辽金近似法将有界增广系统转换成有限维系统; 接着对等价有限维系统进行干扰解耦分析, 推导出系统可干扰解耦的充分条件; 最后通过数值分析和仿真图例表明了所给出条件的有效性.

     

含未知参数的一维人群动态系统的自适应边界控制

针对一类存在扰动的一维人群动态系统,在扩散系数及边界条件系数未知的情况下,设计自适应边界控制律来控制人群向设定的方向平稳疏散.借助李雅普诺夫稳态判据对自适应边界控制律作用下的人群动态系统的稳定性给出了详细的证明.系统的建模及稳定性的证明均在分布参数系统的范畴内完成,避免了模型降阶方法引起的误差的产生.通过一个仿真实例,对比人群动态系统在未施加外部控制, Robin边界控制及自适应边界控制三种情况下,当扩散系数取不同数值时,人群密度的演化情况,验证了自适应边界控制律的有效性.     

分布参数系统边界控制的Haar小波算法

由于分布参数系统通常由偏微分方程描述,采用解析法求解分布参数系统最优边界控制问题,是非常难以解决的.正交函数逼近的方法在分布参数系统控制方面,已经取得了较好的效果.Haar小波作为正交基函数,利用小波的一些运算及变换矩阵,将分布参数系统转化为集总参数系统,再求其逼近解.仿真示例验证了所提出的算法是非常有效的.该方法为分布参数系统的控制算法提出了一条新的解决方案.     

Iterative Learning Control for Distributed Parameter Systems Based on Non-Collocated Sensors and Actuators

In this paper, an open-loop PD-type iterative learning control (ILC) scheme is first proposed for two kinds of distributed parameter systems (DPSs) which are described by parabolic partial differential equations using non-collocated sensors and actuators. Then, a closed-loop PD-type ILC algorithm is extended to a class of distributed parameter systems with a non-collocated single sensor and m actuators when the initial states of the system exist some errors. Under some given assumptions, the convergence conditions of output errors for the systems can be obtained. Finally, one numerical example for a distributed parameter system with a single sensor and two actuators is presented to illustrate the effectiveness of the proposed ILC schemes.      

Model reduction and controller design for a nonlinear heat conduction problem using finite element method

The mathematical models for dynamic distributed parameter systems are given by systems of partial differential equations. With nonlinear material properties, the corresponding finite element (FE) models are large systems of nonlinear ordinary differential equations. However, in most cases, the actual dynamics of interest involve only a few of the variables, for which model reduction strategies based on system theoretical concepts can be immensely useful. This paper considers the problem of controlling a three dimensional profile on nontrivial geometries. Dynamic model is obtained by discretizing the domain using FE method. A nonlinear control law is proposed which transfers any arbitrary initial temperature profile to another arbitrary desired one. The large dynamic model is reduced using proper orthogonal decomposition (POD). Finally, the stability of the control law is proved through Lyapunov analysis. Results of numerical implementation are presented and possible further extensions are identified.     

Automatically determining symmetries of partial differential equations

A REDUCE package for determining the group of Lie symmetries of an arbitrary system of partial differential equations is described. It may be used both interactively and in a batch mode. In many cases the system finds the full group completely automatically. In some other cases there are a few linear differential equations of the determining system left the solution of which cannot be found automatically at present. If it is provided by the user, the infinitesimal generators of the symmetry group are returned.     

Control theory for stochastic distributed parameter systems,an engineering perspective

The main purpose of this paper is to survey some recent progresses on control theory for stochastic distributed parameter systems, i.e., systems governed by stochastic differential equations in infinite dimensions, typically by stochastic partial differential equations. We will explain the new phenomenon and difficulties in the study of controllability and optimal control problems for one dimensional stochastic parabolic equations and stochastic hyperbolic equations. In particular, we shall see that both the formulation of corresponding stochastic control problems and the tools to solve them may differ considerably from their deterministic/finite-dimensional counterparts. More importantly, one has to develop new tools, say, the stochastic transposition method introduced in our previous works, to solve some problems in this field.     

Adaptive fault‐tolerant control for a nonlinear flexible aircraft wing system

Fault‐tolerant control problems have been extensively studied in all kinds of control systems. However, there is little work on fault‐tolerant control for distributed parameter systems. In this paper, a novel adaptive fault‐tolerant boundary control scheme is proposed for a nonlinear flexible aircraft wing system against actuator faults. The whole system is regarded as a distributed parameter system, and the dynamic model of the flexible wing system is described by a set of partial differential equations (PDEs) and ordinary differential equations (ODEs). The proposed controller is designed by using the Lyapunov's direct method and adaptive control strategies. Based on the online estimation of actuator faults, the adaptive controller parameters can update automatically to compensate the actuator faults of the system. Besides, a fault‐tolerant controller is also developed for this system in the presence of external disturbances. Differing from existing works about adaptive fault‐tolerant control, the adaptive controller presented in this paper is designed for a distributed parameter system. Finally, numerical simulations are carried out to illustrate the effectiveness of the proposed control scheme.     

Iterative learning control for hyperbolic distributed parameter systems based on sensor–actuator networks

This paper proposes two kinds of iterative learning control (ILC) schemes for a class of the distributed parameter systems based on sensor–actuator networks which can be described by hyperbolic partial differential equations. A D-type ILC algorithm is first considered and the convergent condition of the output error is obtained via the contraction mapping methodology. Then, the PD-type ILC algorithm is considered in this hyperbolic distributed parameter systems based on sensor–actuator networks. Finally, a cable equation with air and structural damping is given to illustrate the effectiveness of the proposed methods.     

Averaged control

We analyze the problem of controlling parameter-dependent systems. We introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter.First we consider the problem of controllability for linear finite-dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition, in the spirit of the classical rank condition for linear control systems, but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action.We also describe some open problems and directions of possible research, in particular on the average controllability of evolution partial differential equations. In this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system.