The role of controllability and observability in partial pole placement by the method of receptances

In classical control the concept of controllability is used to ensure that the control has the flexibility to change the entire dynamics of the system without excessive control effort. In vibration control the designer is usually confronted with the scenario that only a small number of poles are required to be relocated away from some resonance frequencies. With full controllability the system may suffer from "spill-over" of poles which are not intended to be relocated. Partial pole placement exploits the concept of partial controllability, where the system is controllable only with respect to the desired relocated poles. This principle allows partial pole placement by state feedback control without specifying of the immovable eigenpairs. However, an analytical model of the system has to be available. The method of receptances was developed to avoid using analytical modeling. Measured modal data are used instead of finite element stiffness and mass matrices of large dimensions. The original formulation which used the Sherman-Morrison formula for matrix rank-one modification was applicable only to the single input control. Moreover, the method could not handle partial pole placement since the receptances do not exist at the immovable poles of the open loop system. The concept of observability was used in the recent direct reformulation of the method without the use of the Sherman-Morrison formula. Invariance of poles was achieved by virtue of designed unobservability. Additionally, straightforward extension to the multi-input control was achieved. Also, the same theory of receptance-based partial pole placement by virtue of partial observability was derived in the perspective of continuous systems, which provides further support that the method of receptances is based on the response model of continuous structures obtained from measured experimental data, requiring no discrete or continuous analytical models, model reduction or state estimation.