控制系统中的分形

Fractal Sets in Control Systems

本文将整数维与分形的Hausdorff测度引入并应用于控制系统,同时也介绍了准自相似集这个新概念,证明了这种集合的存在性与唯一性.并将计算自相似集维数的公式推广到准自相似集,在此基础上,说明了控制系统的可达集可以具有分数维.表明在分析非线性系统可控性与可观性时,分形几何学也将是一种有意义的工具.

In this paper the Hausdorff measure of sets of integral and fractional dimensions is introduced and applied to control systems. A new concept, namely,pseudo-self-similar set is also introduced. The existence and uniqueness of such sets are then proved,and the formula for calculating the dimension of self-similar sets is extended to the psuedo-self-similar case. Using the previous theorem, we show that the reachable set of a control system may have fractional dimensions. We hope that as a new approach the geometry of fractal sets will be a proper tool to analyze the controllability and observability of nonlinear systems.