Research Initiation: Structural Properties of Nonlinear Systems with Applications to Their Control

系统模型基于物理原理、定律和简化的假设。这些模型通常具有特定的结构特征，可以在分析系统时加以利用。本研究考虑了其中两个特征。第一种是研究具有对称性的系统。经典的对称概念在哈密顿系统中被扩展到有输入的系统，这是控制工程师感兴趣的。这项研究将进一步提供关于对称系统结构的信息，并将直接利用它来解决某些优化问题。第二个主题是研究离散时间系统的几何基础，重点是研究这类系统的不变分布。这些分布与子系统的经典概念密切相关，并导致系统的局部分解。因此，它们为研究非线性离散时间系统中的各种解耦问题提供了有力的工具。几何方法一直应用于获取非线性系统的结构信息，其特征是存在对称性或不变分布。然后利用这些信息来解决连续和离散时间域中的某些控制问题。

System models are based on physical principles, laws, and simplifying assumptions. These models often possess specific structural characteristics which could be exploited when the system is analyzed. Two of these characteristics are considered in this research. The first concerns the study of systems which possess symmetries. The classical notion of symmetry in Hamiltonian systems is extended to systems with inputs, which are of interest to control engineers. This research will provide further information on the structure of systems with symmetries and will make direct use of this in resolving certain optimization problems. The second topic deals with a geometric basis for the study of discrete time systems and focuses on the study of invariant distributions for such a class of systems. These distributions are closely linked to the classical notion of a subsystem and induce local decompositions of the system. As such, they give a powerful tool for the study of various decoupling problems in nonlinear discrete time systems. Geometric methods are applied consistently to obtain structural information about nonlinear systems, as characterized by the presence of symmetries or invariant distributions. This information is then used to resolve certain control problems in both the continuous and discrete time domains.