Output feedback stabilization of linear port-Hamiltonian descriptor systems

Abstract: This paper presents a structure-preserving method for the stabilization of linear port-Hamiltonian (pH) descriptor systems via output feedback. The stabilization problem is NP-hard for general descriptor systems. Existing approaches often rely on explicit knowledge of the structure-defining matrix $Q$, which is difficult to determine in practice. When $Q$ is unknown, we derive necessary and sufficient conditions under which proportional output feedback ensures that the closed-loop system is regular, impulse-free, asymptotically stable, and retains the port-Hamiltonian structure. These conditions also allow any positive definite matrix to serve as the feedback matrix. The framework is further extended to proportional and derivative output feedback, enabling the assignment of a desired dynamical order. The proposed method thus generalizes existing stabilization results from the special case $Q = I$ to systems with an unknown $Q$, offering a systematic method to structure-preserving stabilization of pH descriptor systems.