On the invariant factors and the minimal polynomial assignments of linear systems (Q6550228)

The article presents a rigorous exploration of the mathematical challenges associated with the control of multiple-input linear systems. It delves into the problems of invariant factors assignment and minimal polynomial assignment, both crucial in the field of control theory.\N\NThe authors begin by revisiting the foundational concepts related to the eigenvalue and eigenstructure assignment problems, highlighting the increased complexity when dealing with multiple-input systems. They identify the less explored areas of invariant factors and minimal polynomial assignments, emphasizing their importance in determining the transient response and overall performance of closed-loop systems.\N\NThe paper's core contribution is the establishment of necessary and sufficient conditions for the solvability of these assignment problems. The authors show that the invariant factors assignment is feasible under certain conditions related to the geometric multiplicity of eigenvalues and the controllability indices of the system. Additionally, they provide a clear link between the system structure and the capacity to assign invariant factors, a relationship that has significant implications for system design and control.\N\NFurthermore, the article introduces a new perspective on the minimal polynomial assignment problem, proposing a necessary and sufficient condition for its solvability. The inclusion of three numerical examples effectively illustrates the theoretical results, providing concrete evidence of the practical applicability of the proposed solutions.\N\NOverall, the article is a significant contribution to the field of control theory, particularly in the area of linear systems. The mathematical rigor and clarity of the arguments make it a valuable resource for researchers and practitioners interested in advanced control system design. However, due to the highly technical nature of the content, the paper may be best suited for readers with a strong background in linear algebra and control theory.