A proof of the minimal order observer

In [8] it is conjectured that the minimal order observer is
p = n - r_{1}
, for a system with r
1
noise-free measurements, r
2
noisy measurements, and
r = r_{1} + r_{2}
. A proof of the existence of the minimal order observer with
p = n - r_{1}
is given in this note. Furthermore, it is shown that from this minimal order observer one can derive the familiar Luenberger observer and Kalman filter as two special cases.