HUANG Tian, XIAO Zhihua, QI Zhenzhong
Firstly, the port-Hamiltonian differential-algebraic systems are transformed into the port-Hamiltonian ordinary differential systems with parameter $\varepsilon$. Then, based on the parameteric ordinary differential systems, two structure-preserving model reduction methods are proposed. The first method is parametric moments matching: Constructing the parametric moments based on the frequency parameter $s$ and the embedding parameter $\varepsilon$ of the parametric systems, and then obtaining the reduced-order models of the parametric systems through parametric moments matching. The reduced-order systems match the parametric moments of the original systems. Finally, by taking the embedded parameter $\varepsilon = 0$, the structure preserving reduced-order models of the original port-Hamiltonian differential-algebraic systems are obtained. The second method is low-rank balanced truncation: Using Laguerre functions to construct the low-rank decomposition factors of the controllability and observability Gramians of the parametric ordinary differential systems. The approximate balanced systems are obtained through projection, and finally, the reduced-order models are constructed by truncating the states corresponding to smaller Hankel singular values. This procedure offers adaptability and enables the construction of reduced-order models meeting specified accuracy conditions while maintaining lower computational complexity. Both algorithms use Gram-Schmidt process to construct new projection matrices, thereby preserving the differential structure of the original system. Finally, the effectiveness of the algorithms is demonstrated through a numerical example.