Designing SISO observers with unknown input

In this paper we investigate the observer design for Single-Input Single-Output (SISO) systems with unknown input. We find that the design conditions proposed by several authors in the past lead to unity relative degree of the plant and zeros of the plant in the open left half of the complex plane, for SISO systems. In other words an observer for a SISO plant with unknown input can be designed if and only if the plant is of unity relative degree and the numerator of the transfer function is Hurwitz. These conditions have an interesting consequence in the context of model reference adaptive control (MRAC) for the case of relative degree one, in the sense that strictly real positive (SPR) condition for the model reference is not needed.