# zbMATH — the first resource for mathematics

Descriptor systems: Decomposition into forward and backward subsystems. (English) Zbl 0534.93013
This paper discusses the geometric structure of the linear discrete-time invariant dynamical system $$Ex_{k+1}=Ax_ k+Bu_ k$$ where E, A are $$n\times n$$ matrices and B is $$n\times n$$. The pencil (E,A) is assumed regular. The key concept is a deflating subspace S. A deflating subspace S is one so that $$\dim \quad(ES+AS)=\dim \quad S$$. Let $$\hat E=(\lambda E+A)^{-1}A$$ where $$\lambda$$ is chosen so that $$\lambda E+A$$ is nonsingular. Note that the deflating subspaces of (E,A) are precisely the invariant subspaces of $$\hat E$$ (this is implicit but not explicit in the paper). The paper presents a unifying geometric framework for known facts about descriptor systems.
Reviewer: S.L.Campbell

##### MSC:
 93B25 Algebraic methods 15A21 Canonical forms, reductions, classification 34A99 General theory for ordinary differential equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 93C05 Linear systems in control theory 93C99 Model systems in control theory 93C55 Discrete-time control/observation systems 47A15 Invariant subspaces of linear operators
##### Keywords:
singular system; descriptor system; deflating subspace
Full Text: