On the Practically Reachable Subspace of the Discrete-Time Systems

Abstract

Spanning vectors of the practically reachable subspace of a single input discrete-time system are studied. Reachable subspace is the space spanned by the vectors of the form b, Ab, A²b, ・・・ , If Aᵏb lies on the subspace Sₖ spanned by the vectors b, Ab, Aᵃb, ・・・ , Aᵏ⁻¹, then Aᵏ⁺lb also lies on the same subspace Sₖ for any l≥0, and so, Sₖ is the reachable subspace. On checking, if Aᵏb lies on Sₖ by numerical calculation, we usually assume that Aᵏb lies on Sₖ in practice, if the distance δ of Aᵏb from Sₖ is small. As the distance of Aᵏ⁺lᵇ from Sₖ is not guaranteed to be small in this case, it must be examined if Aᵏ⁺lb can be assumed to lie on Sₖ, in practice or not. Concerning the distance of Aᵏ⁺l from Sₖ, the following results are obtained, i) When k<n-1, where n is the dimension of the state, Aᵏ⁺lb can have an arbitrary value for n-k-l>l>1. ii) If the maximal number of practically independent vectors are taken from the set of vectors b, Ab, ・・・ , Aⁿ⁻1b, and if the absolute values of the eigenvalues are less than unity, then the distance between Aᵏ⁺lb and the subspace S spanned by these practically independent vectors does not become larger than a certain value MS for any l. The condition that S becomes practically reachable subspace is also studied.