The Method of Receptances for Continuous Rods

It is shown that it is possible in general to assign $2n$ poles of an axially vibrating nonuniform rod by sensing the state at $n$ points along the rod and by using an actuator which applies the control force. Moreover, the assignment of poles may be done without continuous or discrete analytical modeling. In particular, the rigidity, the density, and the variable cross-sectional area of the rod may be considered unknown. The control gains are determined by the receptances between the point of actuation and the points of sensing, which may be measured experimentally. In the analytical arena, where the receptances are exact, the assignment of the desired poles is also exact and suffers from no discretization or model reduction errors. However, the control force affects in general the distribution of the poles which are not intended to be desirably placed or to remain unchanged. Although the analysis is carried out for simplicity on an axially vibrating rod, the above mentioned results are generally applicable to other linear elastic structures of higher dimensions.