Can a fast disturbance observer work under unmodeled actuators?

In this paper, we study robust stability of a closed-loop system with disturbance observer (DOB) when the relative degree of the plant P is greater than or equal to one, but is unknown to the control designer so that he/she chooses the nominal plant P_n of relative degree one. This situation typically happens when the actuator dynamics is ignored at the control design stage. Based on the standing assumption that the bandwidth of the low-pass Q-filter is sufficiently large, our study reveals the following results: 1) if the relative degree of P is equal to, or bigger than the relative degree of P_n by one, robust stabilization may be achieved by carefully designing the Q-filter and the nominal model P_n. It is emphasized that, when the relative degrees are not the same, not only the Q-filter but also the nominal model P_n should be carefully chosen to achieve the goal. 2) When the difference of relative degrees is greater than one, robust stabilization is not possible no matter how Q-filter and P_n are chosen. Therefore, in order to achieve robust stabilization, it is suggested to estimate the relative degree of the plant as close as possible.