Many practical applications give rise to state-space systems of the form
Once more, the state feedback control law
The concepts of controllability and observability of the system can be defined in terms of the associated standard state-space system.
The analogue of the Sylvester-observer equation for the generalized system is
The Luenberger-observer for the generalized system is the same as that of the standard system. That is, it is given by a system of differential equations
It can be shown that, if is a stable matrix, then
approaches zero as time
increases.
If a full-order observer is constructed (), then an estimate
to the state vector
is obtained by solving the system
However, if a reduced-order observer is constructed (), an estimate
of the state vector
is obtained by solving the
system