  This is the second part of four series papers, aiming at the problem of
sensor dynamics compensation for abstract linear systems. Two major issues are
addressed. The first one is about the sensor dynamics compensation in system
observation and the second one is on the disturbance dynamics compensation in
output regulation for linear system. Both of them can be described by the
problem of state observation for an abstract cascade system. We consider these
two apparently different problems from the same abstract linear system point of
view. A new scheme of the observer design for the abstract cascade system is
developed and the exponential convergence of the observation error is
established. It is shown that the error based observer design in the problem of
output regulation can be converted into a sensor dynamics compensation problem
by the well known regulator equations. As a result, a tracking error based
observer for output regulation problem is designed by exploiting the developed
method. As applications, the ordinary differential equations (ODEs) with output
time-delay and an unstable heat equation with ODE sensor dynamics are fully
investigated to validate the theoretical results. The numerical simulations for
the unstable heat system are carried out to validate the proposed method
visually.
