Parameterizing eigenvector chains for eigenstructure assignment in geometric control theory

We consider the problem of assigning the closed-loop eigenstructure of linear time-invariant (LTI) systems using geometric control techniques. Specifically, we focus on the parameterization and construction of feedback matrices that enforce a prescribed eigenstructure. Building on the framework of controlled and conditioned invariant subspaces, we introduce a recursive procedure for computing valid chains of generalized eigenvectors compatible with the system structure, allowing for the complete parameterization of the space of such feedback matrices. Our method identifies and exploits algebraic and geometric constraints intrinsic to the friend construction process. We formulate a numerically stable algorithm that iteratively constructs the kernel spaces associated with the Rosenbrock system matrix, using orthogonal decompositions. The resulting parameterization provides a complete characterization of the admissible eigenstructures under output-nulling constraints, including both internal and external eigenvalues. Once an admissible eigenstructure is selected, the aforementioned recursive procedure can be used to parameterize the space of feedback matrices that enforce it. Our framework enables systematic exploration of feedback designs that satisfy control specifications. This work advances the computational tools available for eigenstructure assignment and opens avenues for integrating such methods into robust and optimal control design strategies.