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

9813284德拉库诺夫本研究项目的目标是开发能够成功处理分布参数系统(DPS)的稳健且在计算上可行的控制方案。控制DPS的主要困难是复杂和强不确定性，例如，模拟弧焊温度场演变的扩散方程，或模拟柔性机械臂振动的Timoshenko方程。尽管为了模拟的目的，人们可以假设加热(或冷却)材料的几何形状和性质是已知的，或者可以忽略机械手的有效载荷，但在实践中，它们在很大范围内变化。通过(1)选择描述密切相关问题的一组适当的边界条件；(2)求解相关的特征值/特征函数问题；(3)调用假设模式方法，获得DPS的可接受模型。其结果是一个无限解耦的常微分方程组，用于所谓的系统模式。在实践中，一个人截断模型，并使用有限数量的方程完成控制设计。为了在总体上验证这种方法，需要检查未建模动态的影响，即控制和观测溢出。另一方面，对于有限维系统，滑模控制方法已经得到了很好的发展，但对于无限维系统，文献中出现的相对较少。这种方法的吸引力在于它对匹配的扰动和参数变化的鲁棒性。该设计思想基于以下步骤：为了解决诸如稳定焊缝宽度或热熔深等控制问题，将目标表示为定义期望流形的系统状态的某一函数。找到一种控制，使得该关系为真的状态空间中的集合形成滑动流形，即在有限时间内可达的积分流形。一旦进入滑动模式，通过选择切换面来规定动态行为，并且系统固有地对系统输入通道中隐含的那类参数变化和外部干扰--所谓的匹配不确定性--不敏感。具有滑动模式的系统对参数变化和扰动的固有不敏感性消除了对精确建模的需要--这是一种非常理想的特性，具有非常显著的理论和实际意义。在本研究项目中，该方法的推广具有数学严谨性，以进一步研究和综合DPS的有效控制和传感算法。***

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. ***