Invariant factor assignment by non-redundant parametric state feedback

This paper investigates the non-redundant parametrization of state feedback gains that assign a prescribed set of invariant factors to a controllable linear time-invariant system. While traditional approaches often rely on Jordan forms, we specify the desired closed-loop structure directly through a chain of monic polynomials and its associated block companion matrix. By exploiting a right-coprime factorization of the input-to-state transfer matrix, we derive a complete but redundant parametrization of all feedback gains achieving the assignment based on the Sylvester equation. In the resulting Sylvester coordinates, two parameter matrices generate the same feedback gain exactly when they differ by the right action of the invertible centralizer of the target block companion matrix. This identifies the non-redundant parameter space with the quotient of the Sylvester parameter space by this action. An explicit formula for the centralizer is derived and used to construct local normalizations for feedback computation. We also characterize the invariant-factor chains that maximize the dimension of the quotient parameter space for a given pole multiset and quantify the trade-off between design freedom and the maximal Jordan-block sizes of the closed-loop matrix. Numerical examples illustrate the theoretical results.