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 假设。 这些模型通常具有特定的结构 在分析系统时可以利用的特性。 本研究考虑了其中的两个特征。 的 首先是研究具有对称性的系统。 的 哈密顿系统中对称性的经典概念被扩展到 系统的输入，这是感兴趣的控制工程师。 这 研究将提供有关系统结构的进一步信息 并将直接利用这一点来解决某些 优化问题 第二个主题涉及几何基础 离散时间系统的研究，并侧重于研究 这样一类系统的不变分布。 这些 分布与经典的子系统概念密切相关 并诱导系统的局部分解。 因此，他们给了一个 研究非线性系统中各种解耦问题的有力工具 离散时间系统 几何方法始终适用于 获得关于非线性系统的结构信息，其特征在于 对称性或不变分布的存在。 这 信息然后被用来解决在两者中的某些控制问题， 连续和离散时间域。

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.