控制系统中的分形

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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).