Circulation Design for Eigenvalue Replacement: Minimizing Eigenvalue Sensitivities

This article first studies the problem of replacing one single real open-loop eigenvalue in a multivariable linear system by state feedback, simultaneously achieving minimization of the sensitivity of this assigned eigenvalue. Two types of sensitivity indices of the assigned closed-loop eigenvalue corresponding, respectively, to the cases of structured and unstructured parameter perturbations are considered. It is shown that for this problem simple and neat analytical globally optimal solutions exist. By using the derived globally optimal solutions repeatedly, this article second proposes a circulation design for eigenstructure assignment in a stabilizable linear system via state feedback with low closed-loop eigenvalue sensitivities. As a consequence, in those rounds of replacing a real open-loop eigenvalue of order 1, the sensitivity index of the assigned closed-loop eigenvalue can be globally minimized. In the case that all the open-loop eigenvalues to be replaced are real ones of order 1, the proposed circulation design not only turns out to be extremely simple and efficient, but also possesses good numerical reliability because it removes completely matrix inverse operations. Two illustrative examples demonstrate the simplicity and effect of the proposed approach.