Research Initiation Award: Local Smooth Stabilization on Nonlinear Systems

随着系统变得复杂，利用非线性模型进行控制系统设计的重要性变得越来越普遍，简化线性模型假设不足以达到控制的目的。使用非线性模型的一个固有问题是，稳定性等性质不容易表征。更困难的是非线性系统的稳定性问题。本文研究了光滑连续时间非线性系统的局部光滑镇定问题。本研究的主要目的是得到一个适用于所有光滑非线性系统的局部光滑镇定（或检验局部光滑镇定）的一般理论。根据所涉及的系统的结构，本研究将在系统地解决稳定问题的算法构建中达到高潮。本研究的进一步扩展可能涉及将这些算法应用于光滑非线性系统的自适应控制。所涉及的主要工具是构造Lyapunov函数，该函数通过使用多元幂级数和分岔分析的爆破方法来检验光滑非线性系统的局部光滑稳定性。该研究将对非线性系统的局部稳定性和稳定性理论做出重要贡献。这项研究对非线性系统控制的应用将产生深远的影响。这一理论潜在应用的例子可以在机械系统的控制中找到，如飞机或机器人的应用。

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.