AFS结构上矩阵的幂等指数估计

Estimation of the idempotent index based on AFS structures

在应用AFS结构(M,τ,X)研究故障诊断问题中,需要寻找正整数r使其满足M2rτ=Mrτ.由于复杂系统对应的AFS结构上矩阵Mτ的阶数较大,为了减少计算量,需要估计出最小的r.就此问题,给出了基于集合M上的布尔矩阵的概念并得出其传递闭包的相关性质,进而在集合M上的布尔矩阵与(0,1)布尔矩阵之间建立一种同态映射并给出其证明,最后应用该映射对r的范围进行了估计.

We need to find r such that M^(2r)_τ=M^r_τ about problems of hitch diagnose with the AFS structure (M,τ,X). But the index of Boolean matrix M_τ in complex systems is usually of very high dimension. We have to estimate the minimal r to reduce complexity in theoretical studies and practical application. In this paper, we introduce the concept of Boolean Matrix on the set M, and discuss some of its correlated properties of transitive closure. We also present a homomorphic mapping between the Boolean Matrix on the set M and (0,1) Boolean Matrix, and prove this theorem. Finally we give the result about the estimate of integer r.