On the input and output reducibility of multivariable linear systems

By introducing into a constant linear system (
F, G, H
) with input vector
u
and output vector
y
an open-loop control
u = Pv
and observer
z = Qy
, a new constant linear system (
F, GP, QH
) results which has input vector
\upsilon
and output vector
z
. The problem investigated is one of constructing (
F, GP, QH
) so that
\upsilon
and
z
have minimal dimension, subject to the condition that the controllability and observability properties of (
F, G, H
) are preserved. It is shown that when the scalar field
F
(over which the system is defined) is infinite, the minimal dimensions of
\upsilon
and
z
are essentially independent of the specific values of the input and output matrices
G
and
H
. It is also shown that this is not the case when
F
is finite. Furthermore, an algorithm is presented for the construction of the minimal input (minimal output) (
F, GP, QH
), which is directly represented in a useful canonical form.