Mathematical Sciences: Global Decompositions of Nonlinear Control Systems

将一个大的非线性控制系统分解成一个相互关联的小尺度的子系统家族是一个自然而然的问题，受到了许多研究者的关注。迄今为止，关于这一问题的绝大多数工作都集中在局部分解上。所谓分解，通常是指变量的改变，从而产生一个等价的系统，该系统至少有两个独立但相互关联的部分。从实用的角度来看，分解是非常重要的，因为它简化了复杂系统的分析。研究人员提出了全球系统分解的概念，这从概念的角度来看是合理的，并有可能产生可应用于具体例子的结果。他将尝试将有关局部分解的已知结果推广到全球环境。我们将寻求全局分解存在的充要条件。对与控制系统相关的对称群的研究将构成本研究的一部分。还将继续将全局分解应用于研究非线性控制系统的全局能控性。本研究试图进一步发展微分几何在控制系统中的应用。如果成功，它可能会对控制理论的应用产生相当重要的结果，如机器人操作器和直升机自动驾驶仪。

The problem of decomposing a large-scale nonlinear control system into an interconnected family of smaller-scale subsystems is a natural one and has received the attention of many researchers. The vast majority of the work done on this problem thus far has concentrated on local decompositions. By decomposition one usually means a change of variables that produces an equivalent system that has at least two independent but interconnected parts. Decomposition is very important from a practical point of view, because it simplifies the analysis of complex systems. The investigator proposes a notion of global system decomposition which is reasonable from a conceptual point of view and has potential of producing results that can be applied to specific examples. He will attempt to extend known results on local decompositions to a global setting. Both necessary and sufficient conditions of existence of global decompositions will be sought. A study of symmetry groups associated with a control system will constitute a portion of this research. Applications of global decompositons to the study of global controllability of nonlinear control systems will also be pursued. This research attempts to develop further the applications of differential geometry to control systems. If succesful, it could produce results of considerable importance for such applications of control theory as robot manipulators and helicopter autopilots.