Topics in Stabilization of Nonlinear Control Systems

由非线性微分方程建模的系统控制是一个重要的研究课题，因为这些模型在制造业、化学过程、汽车、电力和航空航天工业中很普遍。范式技术已成功地应用于简化非线性常微分方程的状态空间表达式。本文提出用范式技术研究非线性控制系统的镇定问题。因此，我们感兴趣的是线性化至少具有一个不可控临界模态的系统。从一些初步的分析可以推断，用范式作为对待分析系统降维的一种方法有很大的优势。我们建议详细研究这种尺寸优势，因为它可以应用于各种问题。一个令人感兴趣的问题是理解和描述非线性控制系统的临界模态和稳定模态的解耦。提出的另一条研究路线是研究与非线性系统相关的李雅普诺夫函数的范式特征。我们也可以用正规形式估计非线性系统的稳定性吸引区域。我们将研究范式和中心流形技术之间的关系，中心流形技术以前在稳定分析中使用过。我们还将研究全反馈和部分反馈线性化，以及近似反馈线性化，看看是否可以更简单地使用范式来实现这些目标。时间尺度技术已应用于控制系统，以简化分析和获得稳定的结果。本文提出的进一步研究方向将是更深入地理解稳定性与时间尺度之间的关系，以及时间尺度技术与非线性控制系统稳定中现有中心流形和范式技术的相关性。＊＊＊

9411145 Schwartz The control of systems modeled by nonlinear differential equations is an important topic to study because of the prevalence of these models in manufacturing, chemical process, automotive, power, and aerospace industries. Normal form techniques have been successfully applied to simplify the state space expression of nonlinear ordinary differential equations. Here we propose to study the stabilization of nonlinear control systems using normal form techniques. Thus, we are interested in systems whose linearizations possess at least one uncontrollable critical mode. From some preliminary analysis, it can be inferred that there is a great advantage to using the normal form as a method to reduce the dimension of the system whose stability is to be analyzed. We propose to study this dimensional advantage in detail as it may be applied to various problems. One question of interest is to understand and characterize the decoupling of the critical and stable modes of a nonlinear control system in normal form. Another line of proposed study is to examine how the normal form characterizes the Lyapunov function associated with the nonlinear system. We would also estimate the region of attraction of stability for nonlinear systems using the normal form. We will study the relationship between the normal form and the center manifold techniques which have been previously used in stabilization analyses. Full and Partial feedback linearization, and approximate feedback linearization will also be studied, to see if these objectives can be achieved more simply using the normal form. Time scale techniques have been applied to control systems in order to simplify analysis and obtain stability results. A further direction of the research herein proposed will be to derive a deeper understanding of the relationship between stability and time scale, and the relevance of time scale techniques to existing center manifold and normal form techniques in stabilization of nonlinear control systems. ***