Circulation Design for Eigenvalue Replacement: Minimizing Condition Numbers

In this article, a circulation design for robust eigenvalue assignment in a stabilizable linear system via state feedback is developed based on a complete parametric partial eigenstructure assignment (ESA) approach. It replaces, in each round, a subset of the open-loop (OL) eigenvalues and minimizes simultaneously the condition number of the closed-loop eigenvector matrix (EVM). It is shown that, in those rounds where a real eigenvalue of order 1 is replaced, the minimization of the condition number can be converted into an equivalent convex minimization problem and thus a globally optimal feedback gain matrix can be obtained. When all the OL eigenvalues to be replaced are real ones of order 1, the proposed circulation design generally turns out to be a very simple method for the entire ESA, which possesses good numerical reliability since matrix inverse operations are completely avoided. The proposed circulation design is effectively demonstrated with an illustrative example.