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