The Adjugate Method: Reassignment with System Classification

Abstract

Further elaborations on the adjugate method for eigenvalue–eigenvector assignment are discussed in this paper. Additional results concerning the admissible pair introduced recently are stated, verified, and commented on. Properties of the adjugate method concerning eigenvalue reassignment of controllable and uncontrollable systems are revealed and pursued further in this study A reformulation of how calculations should be carried out is investigated, pointing out merits, bounds, and limitations. It has been found that reassignment can involve a case where the closed-loop eigenvector companion zi is zi=0 and the associated eligible closed-loop eigenvectors wi are open-loop ones. This can be considered an advantage in the sense of knowing zi and wi beforehand. This also applies to the case of uncontrollable eigenvalue reassignment, where enlarged closed-loop eigenvector subspaces are uncovered, enabling more flexible designs. Assessments of the traditional method compared to the adjugate method are commented on whenever appropriate. Numerical issues are pointed out where Leverrier’s algorithm is adapted for efficient matrix null space and matrix adjugate determination. A feature of the adjugate method is a result where wU=0, indicating that the eigenvalue assigned is uncontrollable. Such a distinctive feature enables a classification procedure for systems regarding controllability and observability. The various concepts pointed out have been demonstrated and authenticated through carefully selected examples.