控制系统中的分形

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名

邮箱

手机号码

标题

留言内容

验证码

Fractal Sets in Control Systems

Lab, of Managmeni, Decision and Information, Academia Sinica, Institute of Systems Science, Academia Sinica, Beijing

摘要: 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.
- 分形 /
- 准自相似 /
- 分数维 /
- 可达集
Abstract: 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.
- fractal set /
- pseudo-self-similar /
- fractional dimension /
- reachable set

[1]	Wonham,M.H.,Linear Multivariable Control:A Geometric Approach,Springer-Verlag,Berlin(1979).
[2]	Hermann,R.and A.J.Krener,Nonlinear controllability and observability,IEEE Trans.Aut.Contr.,AC-22(1977),728-740.
[3]	Falconer,K.J.,The Geometry of Fractal Sets,Cambridge,New York(1985).
[4]	Paladin,G.and A.Vulpiani,Anomalous scaling laws in multifractal objects,Physical Reports(Review Section of Physics Letters),156,4(1987),147-225.
[5]	Thompson,J.M.T.and H.B.Stewart,Nonlinear Dynamics and Chaos,Geometrical Methods for Engineers and Scientists,John Wiley and Sons Ltd.(1986).
[6]	Massey,W.S.,Algebraic Topology:An Introduction,Springer-Verlag(1987).

点击查看大图