A Rosenbrock framework for tangential interpolation of port-Hamiltonian descriptor systems

Figures & data

Figure 1. Sparsity patterns of exemplary full-order matrices $E,A\in {\mathbb{R}}^{n\times n}$ and of the matrices $E_{\mathrm{p}},A_{\mathrm{p}}\in {\mathbb{R}}^{n_2\times n_2}$ of its associated proper subsystem.

Table 1. Proposed tangential interpolation methods for pH-DAEs. The six different categories are derived from the original system’s Kronecker index $\nu$ and whether its transfer function has improper parts.

Category	$\nu$	$n_2$	$n_3$	$n_4/n_1$	$D_{\mathrm{\infty }}$	References
Index-0	$0$	$\ne 0$	$0$	$0$	$0$	[22, Theorem 7]
Index-1	$1$	$\ne 0$	$\ne 0$	$0$	$0$	[13, Theorems 1 and 2]
						[12, Theorem 5]
Proper Index-2	$2$	$\ne 0$	$0$	$\ne 0$	$0$	[13, Theorem 3]
Improper Index-2	$2$	$\ne 0$	$0$	$\ne 0$	$\ne 0$	[13, Theorem 4]
Proper Index-1–2	$2$	$\ne 0$	$\ne 0$	$\ne 0$	$0$	n/a
Improper Index-1–2	$2$	$\ne 0$	$\ne 0$	$\ne 0$	$\ne 0$	n/a

Figure 2. Structure of the RCL ladder networks modelled by RCL-1M and RCL-12S.

Figure 3. Model parameters and sigma plots for the RCL ladder network models RCL-1M and RCL-12S in pH-DAE staircase form.

Figure 4. Reduction of the model RCL-1M to order $r=40$ using different interpolatory MOR methods. Given are the sigma plots of the FOM and ROMs (left) and of the respective error systems (right).

Figure 5. Reduction of the model RCL-12S to different reduced orders $r\in \{2,4,\dots ,20\}$. Plotted are the ${\mathcal{H}}_2$ errors (left) and ${\mathcal{H}}_{\mathrm{\infty }}$ errors (right) for different interpolatory MOR methods.

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