Nonlinear Robust Control and Estimation For Distributed Parameter Systems: A Temperature Field Control Application

这个研究项目的目标是开发鲁棒和计算上可行的控制方案，可以成功地处理分布式参数系统（DPS）。控制DPS的主要困难是复杂性和强不确定性，例如，用扩散方程来模拟弧焊温度场的演变，或者用Timoshenko方程来模拟柔性机械臂的振动。尽管出于模拟目的，人们可以假设加热（或冷却）材料的几何形状和特性是已知的，或者可以忽略机械手有效载荷，但在实践中，它们在很大范围内变化。一个可接受的DPS模型是通过(1)选择一组合适的边界条件来描述一个密切相关的问题；(2)求解相关的特征值/特征函数问题；(3)调用假定模式方法。其结果是一个无限系统的不耦合，常微分方程的所谓的系统模式。在实践中，人们截断模型并使用有限数量的方程来实现控制设计。一般来说，为了验证这种方法，需要检查未建模动力学的影响，即控制和观察溢出。另一方面，存在滑模控制方法，它在有限维系统中发展得很好，但在无限维系统中出现的文献相对较少。该方法的优点是对匹配扰动和参数变化具有鲁棒性。设计思想基于以下步骤：为了解决控制问题，如焊缝宽度或热渗透的稳定，目标被制定为系统状态的某个函数，该函数定义了所需的流形。我们找到了这样一个控制，在状态空间中，这个关系成立的集合形成了一个滑动流形，即在有限时间e内可到达的积分流形。一旦进入滑模，动态行为由切换曲面的选择规定，系统本质上对系统输入通道中隐含的那类参数变化和外部干扰不敏感——即所谓的匹配不确定性。具有滑动模式的系统对参数变化和干扰的固有不敏感消除了精确建模的需要-这是一个非常理想的性质，具有相当显著的理论和实际意义。本课题以数学严谨的方法对该方法进行推广，进一步研究和综合高效的DPS控制和传感算法。＊＊＊

9813284 Drakunov The objective of this research project is to develop robust and computationally feasible control schemes which can successfully deal with Distributed Parameter Systems(DPS). The main difficulties to control DPS for example, a diffusion equation modeling the evolution of the temperature field in arc welding, or a Timoshenko equation modeling the vibrations of a flexible manipulator, are complexity and strong uncertainty. Although one can assume for simulation purposes that the geometry and properties of the heated (or cooled) materials are known, or that the manipulator payload may be ignored, in practice they vary in a wide range. An accepted model for a DPS is obtained by (1) selecting a suitable set of boundary conditions that describe a closely related problem; (2) solving the associated eigenvalue/eigenfuction problem; and (3) invoking the assumed-mode method. The result is an infinite system of uncoupled, ordinary differential equations for the so called system modes. In practice, one truncates the model and fulfills the control design using a finite number of equations. To validate such an approach in general, the effects of unmodeled dynamics, that is, control and observation spillover, need to be examined. On the other hand, there exists the sliding mode control methodology, which is well developed for finite dimensional systems but relatively very little has appeared in the literature for infinite dimensional systems. The appeal of this approach is its robutsness to matched disturbances and parameter variations. The design idea is based on the following procedure: in order to solve a control problem such as the stabilization of the weld width or heat penetration, the objective is formulated as a certain function of the system states which defines a desired manifold. A control is found such that the set in the state space where this relation is true forms a sliding manifold, that is, an integral manifold reachable in finite tim e. Once in the sliding mode, the dynamic behavior is prescribed by the choice of switching surfaces, and the system is inherently insensitive to that class of parameter variations and external disturbances which is implicit in the system input channels -- the so called matched uncertainty. The inherent insensitivity of a system with sliding modes to parameter variations and disturbances eliminates the need for exact modeling - a very desirable property that has quite remarkable theoretical and practical implications. In this research project, the generalization of this method is developed with mathematical rigor to further study and synthesize efficient control and sensing algorithms for DPS. ***