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

9813284 Drakunov本研究项目的目标是开发强大的和计算上可行的控制方案，可以成功地处理分布参数系统（DPS）。 控制DPS的主要困难是复杂性和很强的不确定性，例如，模拟电弧焊接中温度场演变的扩散方程，或模拟柔性机械手的振动的Eklshenko方程。 尽管出于模拟的目的，可以假设加热（或冷却）材料的几何形状和性质是已知的，或者可以忽略操纵器有效载荷，但实际上它们在很宽的范围内变化。 DPS的可接受模型通过以下步骤获得：（1）选择描述密切相关问题的一组合适的边界条件;（2）求解相关的特征值/特征函数问题;以及（3）调用假定模式方法。 其结果是一个无限系统的非耦合，常微分方程的所谓系统模式。 在实践中，一个截断的模型和实现控制设计使用有限数量的方程。 为了验证这种方法，一般来说，未建模的动态，即控制和观察溢出的影响，需要进行检查。 另一方面，存在滑模控制方法，这是很好地发展有限维系统，但相对很少出现在文献中的无限维系统。 这种方法的吸引力在于它对匹配干扰和参数变化的鲁棒性。设计思想是基于以下过程：为了解决控制问题，如稳定的焊缝宽度或热渗透，目标是制定为一个特定的功能的系统状态，它定义了一个所需的流形。 找到一个控制，使得在状态空间中，这个关系是真的集合形成一个滑动流形，也就是说，在有限时间内可达的积分流形。 一旦进入滑动模式，动态行为由切换表面的选择来规定，并且系统固有地对隐含在系统输入通道中的那类参数变化和外部干扰不敏感-所谓的匹配不确定性。 滑模系统对参数变化和干扰的固有不敏感性消除了精确建模的需要-这是一个非常理想的特性，具有相当显着的理论和实际意义。 在本研究计画中，此方法的推广是以数学的严谨性来发展，以进一步研究和综合DPS的有效控制与感测演算法。 ***

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. ***