  In previous work the notion of input to state stability (ISS) has been
generalized to systems with outputs, yielding a number of useful concepts. When
considering a system whose output is to be kept small (i.e. an error output),
the notion of input to output stability (IOS) arises. Alternatively, when
considering a system whose output is meant to provide information about the
state (i.e. a measurement output), one arrives at the detectability notion of
output to state stability (OSS). Combining these concepts, one may consider a
system with two outputs, an error and a measurement. This leads naturally to a
notion of partial detectability we call measurement to error stability (MES).
This property characterizes systems in which the error signal is detectable
through the measurement signal.
  This paper provides a partial Lyapunov characterization of the MES property.
A closely related property of stability in three measures (SIT) is introduced,
which characterizes systems for which the error decays whenever it dominates
the measurement. The SIT property is shown to imply MES, and the two are shown
to be equivalent under an additional boundedness assumption. A nonsmooth
Lyapunov characterization of the SIT property is provided, which yields the
partial characterization of MES. The analysis is carried out on systems
described by differential inclusions -- implicitly incorporating a disturbance
input with compact value-set.
