Mathematical Sciences: Control Structures

小行星9409037 该项目由应用数学计划和几何分析计划共同资助。它的目标是构造和应用新的不变量，结构定理和正常形式的控制系统下的时间无关，时间依赖，和动态反馈。这些工具涉及的方法等效精制的主要研究者，外部微分系统，和理论的无限伪群。一个重要的原则是扩大的概念反馈等效的完全非线性类别。该程序导致了精确线性化算法的重大改进。 类似的想法将被用来研究其他适用的标准形式，如标准双线性系统的算法。一个新的概念，控制结构，一个更全面的观点，非奇异控制系统，也将进行调查。 反馈控制系统的理论，最初是由工程问题激发的，在应用数学中占据中心地位，是控制工程应用的基础。这个项目将开发的几何基础，高性能自动控制系统的动力学。它主要关注具有高度非线性动态的系统，并且可能具有许多相互影响的相互关联的组件。工程要求更高的性能，甚至最优性，往往需要全新的控制设计，因为经典的线性化技术的控制设计往往是不够的，在这种情况下。通过采用纯数学的各个领域的技术，如微分几何，精确线性化算法的重大改进，应用于非线性自动驾驶仪的设计，例如，预计。

9409037 Gardner This project is funded jointly by the Applied Mathematics Program and the Geometric Analysis Program. Its goal is to construct and apply new invariants, structure theorems and normal forms of control systems under time-independent, time-dependent, and dynamic feedback. The tools involve the method of equivalence as refined by the principal investigator, exterior differential systems, and the theory of infinite pseudogroups. An important principle is the enlargement of the notion of feedback equivalence to the fully nonlinear category. This program has resulted in major improvements in exact linearization algorithms. Similar ideas will be used to investigate algorithms for other applicable normal forms such as standard bilinear systems. A new concept of control structure, a more global viewpoint of nonsingular control systems, will also be investigated. The theory of feedback control systems, originally motivated by problems in engineering, occupies a central place in applied mathematics and is fundamental to control engineering applications. This project will develop the geometric foundations that underlie the dynamics of high-performance automatic control systems. It focuses primarily on systems that have highly nonlinear dynamics and which may have many interconnected components influencing each other. The engineering demands for ever higher performance, or even optimality, often require completely new control designs since classical linearization techniques for control design is often inadequate in such situations. By employing techniques from various areas of pure mathematics, such as differential geometry, major improvements in exact linearization algorithms, with applications to nonlinear autopilot design, for example, are expected.