Circulation Design for Eigenvalue Replacement: Minimizing Gain Magnitude

Based on a complete parametric partial eigenstructure assignment (ESA) approach, a circulation design for ESA in a stabilizable linear system via state feedback is proposed. It is shown that, in those rounds where a real open-loop eigenvalue of order 1 is replaced, a type of very general indices can be globally minimized, which include those for general eigenvector configuration and gain magnitude minimization as special cases. When all the open-loop eigenvalues to be replaced are real and are of order 1, the proposed circulation design turns out to be extremely simple and efficient, and also possesses good numerical reliability since the globally optimal solution is established in a simple, direct, and explicit form, and the proposed circulation algorithms remove completely matrix inverse operations. An illustrative example demonstrates the simplicity and effect of the proposed approach.