On spectral controllability of multi-input time-delay systems
The systems under consideration are of the form \(\dot x(t)=A_ 0x(t)+A_ 1x(t-h)+...+A_ Nx(t-Nh)+Bu(t)\) where x(t) is in \(R^ n\) and u(t) in \(R^ m\). The spectral controllability condition, as is well- known, can be expressed as rank(sI-\(\sum^{N}_{k=0}(\exp -skh)A_ k,B)=n\) for all complex s. The question considered is under what assumption B can be substituted by Br, for some vector r, without violating this condition. For systems without delays \((N=0)\) the situation is clear: \(A_ 0\) must be cyclic. For time-delay systems the main theorem in the paper gives the following necessary and sufficient conditions for the existence of a vector r with the required properties: 1) A(z) is cyclic for some complex z, where A(z) is defined as \(A(z)=A_ 0+zA_ 1+...+z^ NA_ N.\) 2) If s is such that \(A(e^{-sh})\) is not cyclic then rank \([sI-A(e^{- sh})]\geq n-1\).
scientific article; zbMATH DE number 3995518
scientific article