On the invariant factors and the minimal polynomial assignments of linear systems

In this note, the invariant factors assignment and the minimal polynomial assignment problems for multiple-input linear systems are studied. It is found that for any invariant factors with the geometry multiplicity of each eigenvalue less than or equal to the number of the inputs, the invariant factors assignment problems are solvable if and only if the controllability indices set of the linear system satisfies certain condition. For the minimal polynomial assignment problem, a necessary and sufficient condition on the solvability is proposed. Three numerical examples are provided to illustrate the proposed results.