Eigenvalue sensitivity minimisation for robust pole placement by the receptance method

The problem of robust pole placement in active structural vibration control by the method of receptance is considered in this paper. Expressions are derived for the eigenvalue sensitivities to parametric perturbations, which are subsequently minimised to improve performance robustness of the control of a dynamical system. The described approach has application to a vibrating system where variations are present due to manufacturing and material tolerances, damages and environment variabilities. The closed-loop eigenvalue sensitivities are expressed as a linear function of the velocity and displacement feedback gains, allowing their minimisation with carefully calculated feedback gains. The proposed algorithm involves curve fitting perturbed frequency response functions, FRFs, using the rational fraction polynomial method and implementation of a polynomial fit to the individual estimated rational fraction coefficients. This allows the eigenvalue sensitivity to be obtained entirely from structural FRFs, which is consistent with the receptance method. This avoids the need to evaluate the M,C,K matrices which are typically obtained through finite element modelling, that produces modelling uncertainty. It is also demonstrated that the sensitivity minimisation technique can work in conjunction with the pole placement and partial pole placement technique using the receptance method. To illustrate the working of the proposed algorithm, the controller is first implemented numerically and then experimentally.