Research Initiation Award: Local Smooth Stabilization on Nonlinear Systems

利用非线性模型的重要性 控制系统设计变得越来越普遍， 系统变得复杂，简化线性模型 假设不足以控制。 使用非线性模型的固有问题是， 像稳定性这样的特性， 表征了 更困难的是 非线性系统的可镇定性 此处为 局部光滑问题的研究 光滑连续时间非线性系统镇定 系统提出。 的主要目标 建议的研究是为了获得一个普遍的理论， 可以应用于局部平滑稳定（或 测试局部光滑稳定性）的所有光滑 非线性系统 这项研究将最终在 构建算法， 解决稳定问题，这取决于 所涉及的系统结构。 进一步延长 这项研究可能涉及使用这些算法， 平滑非线性系统自适应控制。 所涉及的主要工具将是建造 李雅普诺夫函数检验局部光滑 光滑非线性系统通过 利用多元幂级数和blow-up 分叉分析方法 这项研究将为 非线性系统的局部稳定性和可镇定性理论 系统. 这项研究的影响将是深远的 达到应用程序的控制， 非线性系统 潜在应用实例 这一理论可以在机械控制中找到 系统，如飞机或机器人应用。

The importance of utilizing nonlinear models for control systems design has become more prevalent as systems become complex, and simplifying linear model assumptions are inadequate for purposes of control. An inherent problem in using nonlinear models is that properties such as stability are not so easily characterized. Even more difficult is the question of stabilizability of nonlinear systems. Here an investigation of the problem of local smooth stabilization of smooth continuous time nonlinear systems is proposed. The main objective of the proposed research is to obtain a general theory which may apply toward the local smooth stabilization (or testing the local smooth stabilizability) of all smooth nonlinear systems. This research will culminate in the construction of algorithms which will systematically attack the stabilization problem, depending on the structure of the system involved. Further extensions of this research could involve the use of these algorithms in adaptive control of smooth nonlinear systems. The main tools involved will be the construction of Lyapunov functions which test the local smooth stabilizability of smooth nonlinear systems through the use of multivariate power series and the blow-up method of bifurcation analysis. This research will make a significant contribution to the theory of local stability and stabilizability of nonlinear systems. The impact of this research will be far reaching for applications for applications of control of nonlinear systems. Examples of potential applications of this theory can be found in control of mechanical systems such as aircraft or robotic applications.