Partial closed-loop pole assignment via Sylvester equation for linear time-invariant systems

This paper introduces a novel approach of partial closed-loop pole assignment for linear time-invariant (LTI) systems. A linear algebraic condition, derived from the Sylvester equation of the system, is presented to characterize the state feedback gains that allocate a subset of the closed-loop poles to a prescribed set of points. We show that this condition can serve as a constraint for designing controllers to achieve other control objectives, for instance, maintaining the other unspecified poles unchanged or minimizing the $H_{\infty}$ norm of the closed-loop system. Furthermore, a connection between the proposed method and moment-matching-based model reduction is also discussed, which shows that the controller gain can be designed based on a reduced-order model that matches certain moments of the original system. Finally, some numerical examples are used to illustrate the proposed method.