Assignment of eigenvalue sensitivities from receptance measurements

The paper addresses the problem of eigenvalue assignment in active vibration control by the receptance method using single-input state feedback. It is demonstrated that not only can assignment be applied to the poles of the system but also to the sensitivities of the poles. The approach described has applications particularly in vibration absorption and the detuning of structures to avoid resonance. Insensitivity of the poles to perturbations in the system gains is desirable from the point of view of robust dynamical performance. The perturbation method is applied to a characteristic equation resulting in expressions for the derivatives of the pole eigenvalues with respect to the control gains. The method is based entirely upon measured vibration data in the form of receptances, there being no need to determine or to know the M, C, K matrices. The closed-loop sensitivities are expressed as linear functions of the control gains, thereby making accessible the assignment of the sensitivities to chosen values. For the case of the open-loop system, the method is shown to produce identical results to established eigenvalue sensitivity procedures. It is demonstrated that chosen eigenvalues may be rendered insensitive to all the control gain terms by choice of the distribution vector b. Numerical examples are provided to illustrate the working of the technique.