Linear state-variable control laws with reduced-order observers may produce unstable feedback controllers

Prior to 1979, it was a commonly held belief among modern control researchers that a stabilizing linear feedback control law u=Kxˆ, where xˆ is obtained from a state observer, would always produce a stable “controller” u(s)=G(s)γ(s) for any completely controllable/observable linear plant x˙=Ax+Bu. However, in a 1979 paper it was proven that, in fact, the controller subsystem u(s)=G(s)γ(s) of the stabilized closed-loop system x˙=Ax+BKxˆ may be unstable when a full-order observer is used to generate xˆ. It is shown that the same possibility of controller instability also arises for the case when a reduced-order observer is used to generate xˆ