New techniques for the design of observers

In this note we prove that if in observer equation
VA - FV = GC
we choose matrix
F
in companion form, it is possible to give a very simple, explicit solution to unknown matrix
V
. It is furthermore shown that for almost all the matrices
G
, such that the pair (
F, G
) is reachable, matrix
(c\over{v})
is of full rank. All this allows us to establish efficient algorthims for designing reduced observers both for the estimate of state and for that of a given linear functional, even if the spectra of
F
and
A
are not disjointed.