This paper applies the proportional-integral (PI) observer in connecti
on with loop-transfer recovery (LTR) design for discrete-time systems.
Both the prediction and the filtering versions of the PI observer are
considered. We show that a PI observer makes it possible to obtain ti
me recovery, i.e., exact recovery for t --> infinity: under mild condi
tions. Two systematic LTR design methods, one based on an extension of
the linear quadratic Gaussian loop-transfer recovery (LQG/LTR) and th
e other based on linear matrix inequality (LMI), are derived for the d
iscrete-time PI observer case. Explicit expressions for the recovery e
rror when exact recovery is not achievable for all frequencies are als
o given.