Keywords

1 Introduction

Port-Hamiltonian (pH) systems are network-based models that arise within a modeling framework in which a physical system model is decomposed into hierarchies of submodels interconnected principally through the exchange of energy. The submodels typically reflect one of a variety of core modeling paradigms that describe phenomenological aspects of the dynamics having different physical character, such as e.g. electrical, thermodynamic, or mechanical. The pH framework is able to knit together submodels featuring dramatically different physics through a disciplined focus on energy flux as a principal mode of system interconnection. pH structure is inherited via power conserving interconnection and a variety of physical properties and functional constraints (e.g., passivity and energy and momentum conservation) are encoded directly into the structure of the model equations [5, 30]. Interconnection of submodels may create further constraints on system behavior and evolution, originating as conservation laws (e.g., Kirchoff’s laws or mass balance), or as position and velocity limitations in mechanical systems. As a result, system models are often naturally posed as combinations of dynamical system equations and algebraic constraint equations, i.e., as port-Hamiltonian descriptor systems or port-Hamiltonian differential-algebraic equations (pHDAE).

When a pHDAE system is linearized around a stationary solution, one obtains a linear time-invariant pHDAE with specially structured coefficient matrices, see [5]. Although the approach we develop here can be extended easily to more general settings, we narrow our focus to a particular formulation offered as one of the simpler among a variety of formulations presented in [5, 22].

Definition 1

A linear time-invariant DAE system of the form

$$\begin{aligned} \begin{array}{rl} {\mathbf{E}} \dot{ {\mathbf{x}} }\!\!\!\!\!\!&{}=( {\mathbf{J}} - {\mathbf{R}} )\, {\mathbf{x}} +( {\mathbf{B}} - {\mathbf{P}} )\, {\mathbf{u}} ,\\[2mm] {\mathbf{y}} \!\!\!\!\!\!&{}=( {\mathbf{B}} + {\mathbf{P}} )^T\, {\mathbf{x}} +( {\mathbf{S}} + {\mathbf{N}} )\, {\mathbf{u}} , \end{array} \qquad {\mathbf{x}} (t_0)=0, \end{aligned}$$
(1)

with \( {\mathbf{E}} \), \( {\mathbf{J}} \), \( {\mathbf{R}} \in {}\mathsf {I}\!\mathsf {R}^{n \times n}\), \( {\mathbf{B}} , {\mathbf{P}} \in {}\mathsf {I}\!\mathsf {R}^{n \times m}\), \( {\mathbf{S}} = {\mathbf{S}} ^T\), \( {\mathbf{N}} =- {\mathbf{N}} ^T\in {}\mathsf {I}\!\mathsf {R}^{m \times m}\), on a compact interval \(\mathbb I\subset {}\mathsf {I}\!\mathsf {R}\), is a pHDAE system if the following properties are satisfied:

  1. 1.

    The differential-algebraic operator

    $$\begin{aligned} {\mathbf{E}} \frac{d}{dt}- {\mathbf{J}} : {\mathcal C}^1(\mathbb {I},{}\mathsf {I}\!\mathsf {R}^n)\rightarrow {\mathcal C}^0(\mathbb {I},\mathbb {R}^n) \end{aligned}$$

    is skew-adjoint, i.e., \( {\mathbf{J}} ^T=- {\mathbf{J}} \) and \( {\mathbf{E}} = {\mathbf{E}} ^T\),

  2. 2.

    \( {\mathbf{E}} \) is positive semidefinite, i.e., \( {\mathbf{E}} \ge 0\), and

  3. 3.

    the passivity matrix

    $$\begin{aligned} {\mathbf{W}} =\begin{bmatrix} {\mathbf{R}} &{} {\mathbf{P}} \\ {\mathbf{P}} ^T &{} {\mathbf{S}} \end{bmatrix}\in \mathbb {R}^{(n+m)\times (n+m)} \end{aligned}$$

    is symmetric positive semi-definite, i.e., \( {\mathbf{W}} = {\mathbf{W}} ^T\ge 0\).

The Hamiltonian function \({\mathcal H}:\mathbb {R}^n\rightarrow \mathbb {R}\) of the system is \({\mathcal H}( {\mathbf{x}} )=\frac{1}{2} {\mathbf{x}} ^T {\mathbf{E}} {\mathbf{x}} .\) \({\mathcal H}\) is a quadratic function of state describing energy storage within the system. The matrix \( {\mathbf{E}} \), which is the Hessian matrix of \({\mathcal H}( {\mathbf{x}} )\), is the energy matrix; \( {\mathbf{R}} \) is the dissipation matrix; \( {\mathbf{J}} \) is the structure matrix describing the energy flux among internal energy storage elements; and \( {\mathbf{B}} \pm {\mathbf{P}} \) are port matrices describing how energy enters and leaves the system. \( {\mathbf{S}} \) and \( {\mathbf{N}} \) are matrices associated with a direct feed-through from input \( {\mathbf{u}} \) to output \( {\mathbf{y}} \). If the system has a solution \( {\mathbf{x}} \in \mathcal {C}^1(\mathbb {I},\mathbb {R}^n)\) in \(\mathbb {I}\) for a given input function \( {\mathbf{u}} \), then the dissipation inequality

$$\begin{aligned} \frac{d}{dt}{\mathcal H}( {\mathbf{x}} )= {\mathbf{u}} ^T {\mathbf{y}} -\begin{bmatrix} {\mathbf{x}} \\ {\mathbf{u}} \end{bmatrix}^T {\mathbf{W}} \begin{bmatrix} {\mathbf{x}} \\ {\mathbf{u}} \end{bmatrix} \ \le \ {\mathbf{u}} ^T {\mathbf{y}} \end{aligned}$$

must hold. The system is both passive and Lyapunov stable, as \(\mathcal H\) defines a Lyapunov function when \( {\mathbf{u}} =0\), see [5].

In practice, network-based automated modeling tools such as modelica, Matlab/Simulink, or 20-simFootnote 1 may produce system models that are overdetermined as a consequence of redundant modeling. Thus, automated generation of a system model usually must be followed by reformulation and regularization. Evidently, these subsequent steps should preserve the pH structure and retain compatibility with standard simulation, control, and optimization tools.

pHDAE models may be very large and complex, e.g., models that arise from semi-discretized (in space) continuum models in hydrodynamics [11, 19, 29], or mechanics [25]. In such cases, model reduction techniques are necessary to apply control and optimization methods. Preservation of the pH structure, the constraint structure, and the interconnection structure is necessary to maintain model integrity. For linear time-invariant pH systems with positive-definite \( {\mathbf{E}} \), such methods are well-developed, including tangential interpolation approaches [15, 16] and moment matching [26, 28, 31], as well as effort and flow constraint reduction methods [27]. Interpolatory approaches have been extended to nonlinear pH systems as well, see [3] and [9].

In the case of singular \( {\mathbf{E}} \), only recently in [11, 18] have model reduction techniques been introduced that are able to preserve pH structure. Note that for unstructured DAEs, constraint-preserving reduction methods were introduced in [17, 19, 25, 29].

In this work, we discuss particular structure-preserving model reduction methods that incorporate both regularization and interpolation for linear pHDAE systems having the form (1). For different model classes, we describe how constraints are represented in the transfer function and how the polynomial part can be preserved with interpolatory model reduction methods. The key step identifies, as in [5, 24], all redundancies as well as both explicit and implicit system constraints, partitioning the system equations into redundant, algebraic, and dynamic parts. Then, only the dynamic part need be reduced in a way that preserves the structure.

In Sect. 3, we consider pHDAEs of the form (1) and note important simplifications of the general regularization procedure. Structure preserving model reduction of pHDAEs using interpolatory methods is discussed in Sect. 4 for several specific model structures. We propose an algorithm to generate effective interpolation data in Sect. 5 followed by a numerical example that demonstrates its effectiveness.

2 General Differential-Algebraic Systems

In this section, we review some basic properties of linear constant coefficient DAE systems having the general form,

$$\begin{aligned} \begin{array}{rl} {\mathbf{E}} \dot{ {\mathbf{x}} }\!\!\!\! &{}= {\mathbf{A}} {\mathbf{x}} + {\mathbf{B}} {\mathbf{u}} ,\\[2mm] {\mathbf{y}} \!\!\!\! &{}= {\mathbf{C}} {\mathbf{x}} + {\mathbf{D}} {\mathbf{u}} , \end{array} \qquad {\mathbf{x}} (t_0)=0, \end{aligned}$$
(2)

where \( {\mathbf{E}} ,\, {\mathbf{A}} \in {}\mathsf {I}\!\mathsf {R}^{n \times n}\), \( {\mathbf{B}} \in {}\mathsf {I}\!\mathsf {R}^{n \times m}\), \( {\mathbf{C}} \in {}\mathsf {I}\!\mathsf {R}^{p \times n}\), and \( {\mathbf{D}} \in {}\mathsf {I}\!\mathsf {R}^{p \times m}\); for details, see, e.g., [7].

The matrix pencil \(\lambda {\mathbf{E}} - {\mathbf{A}} \) is said to be regular, if \(\det (\lambda {\mathbf{E}} - {\mathbf{A}} ) \ne 0\) for some \(\lambda \in {}\mathsf {I}\!\mathsf {C}\). For regular pencils, the finite eigenvalues are the values \(\lambda \in {}\mathsf {I}\!\mathsf {C}\) for which \(\det (\lambda {\mathbf{E}} - {\mathbf{A}} ) = 0\). If the reversed pencil \(\lambda {\mathbf{A}} - {\mathbf{E}} \) has the eigenvalue 0, then this is the infinite eigenvalue of \(\lambda {\mathbf{E}} - {\mathbf{A}} \).

With zero initial conditions, \( {\mathbf{x}} (t_0)=0\) as in (2), and a regular pencil, \(\lambda {\mathbf{E}} - {\mathbf{A}} \), we obtain in the frequency domain, the transfer function, \( {\mathbf{H}} (s)= {\mathbf{C}} (s {\mathbf{E}} - {\mathbf{A}} )^{-1} {\mathbf{B}} + {\mathbf{D}} \). \( {\mathbf{H}} (s)\) is a rational function mapping the Laplace transform of the input function \( {\mathbf{u}} \) to the Laplace transform of the output function \( {\mathbf{y}} \). If the transfer function is written as

$$ {\mathbf{H}} (s)= {\mathbf{C}} (s {\mathbf{E}} - {\mathbf{A}} )^{-1} {\mathbf{B}} + {\mathbf{D}} = {\mathbf{G}} (s)+ {\mathbf{P}} (s), $$

where \( {\mathbf{G}} (s)\) is a proper rational matrix function and \( {\mathbf{P}} (s)\) is a polynomial matrix function, then the finite eigenvalues are the poles of the proper rational part, \( {\mathbf{G}} (s)\), while the infinite eigenvalues are associated with the polynomial part, \( {\mathbf{P}} (s)\).

Regular pencils can be analyzed via the Weierstraß Canonical Form; see, e.g., [13]. The index, \(\nu \), of the pencil \(\lambda {\mathbf{E}} - {\mathbf{A}} \) is the size of the largest block associated with the eigenvalue \(\infty \) in the Weierstraß canonical form; if \( {\mathbf{E}} \) is nonsingular, then \(\nu =0\).

To analyze general DAEs and to understand the properties of the transfer function, conditions on controllability and observability are needed, see, e.g., [7, 10]. Let \(S_{\infty }( {\mathbf{E}} )\) and \(T_{\infty }( {\mathbf{E}} )\) be two matrices with orthonormal columns spanning, respectively, the right and left nullspace of \( {\mathbf{E}} \). A system is called strongly controllable if it satisfies

\(\mathrm{C1}\!:\ \mathop {\mathrm {rank}}\nolimits [ \lambda {\mathbf{E}} - {\mathbf{A}} ,\, {\mathbf{B}} ] = n \text { for all } \lambda \in {}\mathsf {I}\!\mathsf {C}\)    and     \(\mathrm{C2}\!:\ \mathop {\mathrm {rank}}\nolimits [ {\mathbf{E}} ,\, {\mathbf{A}} S_{\infty }( {\mathbf{M}} ),\, {\mathbf{B}} ] = n\).

Analogously, a system is called strongly observable if it satisfies

\(\mathrm{O1}\!: \ \mathop {\mathrm {rank}}\nolimits \left[ \begin{array}{cc}\lambda {\mathbf{E}} ^T- {\mathbf{A}} ^T&{\mathbf{C}} ^T\end{array}\right] = n \text { for all }\lambda \in {}\mathsf {I}\!\mathsf {C}\) and \(\mathrm{O2}\!: \mathop {\mathrm {rank}}\nolimits \left[ \begin{array}{ccc} {\mathbf{E}} ^T&{\mathbf{A}} ^T T_{\infty }( {\mathbf{E}} )&{\mathbf{C}} ^T \end{array}\right] = n\).

If a system is strongly controllable and strongly observable, then it is called minimal.

Conditions (C1) and (C2) are preserved under non-singular equivalence transformations as well as under state and output feedback. Note, however, that regularity (or non-regularity) of the pencil, of the index, and of the polynomial part of the transfer function are generally not preserved under state or output feedback. For systems that satisfy (C2), there exists a suitable linear state feedback matrix, \( {\mathbf{F}} _1\), such that \(\lambda {\mathbf{E}} -( {\mathbf{A}} + {\mathbf{B}} {\mathbf{F}} _1)\) is regular and of index at most one. Also if conditions (C2) and (O2) hold, then there exists a linear output feedback matrix, \( {\mathbf{F}} _2\), so that the pencil \(\lambda {\mathbf{E}} -( {\mathbf{A}} + {\mathbf{B}} {\mathbf{F}} _2 {\mathbf{C}} )\) has this property, see [7].

3 Regularization of PHDAE Systems

In general, it cannot be guaranteed that a system, generated either from a realization procedure or an automated modeling procedure, has a regular pencil \(\lambda {\mathbf{E}} - {\mathbf{A}} \). Therefore, a regularization procedure typically must be applied, see [6, 8, 20]. For pHDAEs, pH structure helps simplify these regularization procedures significantly, since the pencil \(\lambda {\mathbf{E}} -( {\mathbf{J}} - {\mathbf{R}} )\) is associated with a free dissipative Hamiltonian DAE system (i.e., \( {\mathbf{u}} =0\)). Such systems have many nice properties [22]: The index \(\nu \) is at most two; all eigenvalues are in the closed left half plane; and all eigenvalues on the imaginary axis are semi-simple (except for possibly the eigenvalue 0, which then may have Jordan blocks of size at most two). A singular pencil can occur only when \( {\mathbf{E}} , {\mathbf{J}} \), and \( {\mathbf{R}} \) have a common nullspace [23], so if one is able to compute efficiently this common nullspace, it is possible to remove explicitly the singular part.

Lemma 1

For the pHDAE in (1), there exists an orthogonal basis transformation matrix \( {\mathbf{V}} \in {}\mathsf {I}\!\mathsf {R}^{n \times n}\) such that in the new variable \( {\widetilde{\mathbf{x}}} = \left[ \begin{array}{ccc} {\widetilde{\mathbf{x}}} _1^{\,T}&{\widetilde{\mathbf{x}}} _2^{\,T}&{\widetilde{\mathbf{x}}} _3^{\,T}\end{array}\right] ^T = {\mathbf{V}} ^T {\mathbf{x}} \), the system has the form

$$\begin{aligned} \left[ \begin{array}{ccc} {\mathbf{E}} _1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 \end{array}\right] \left[ \begin{array}{c} \dot{ {\widetilde{\mathbf{x}}} }_1 \\ \dot{ {\widetilde{\mathbf{x}}} }_2 \\ \dot{ {\widetilde{\mathbf{x}}} }_3 \end{array}\right]&= \left[ \begin{array}{ccc} {\mathbf{J}} _1- {\mathbf{R}} _1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0\end{array}\right] \left[ \begin{array}{c} {\widetilde{\mathbf{x}}} _1 \\ {\widetilde{\mathbf{x}}} _2 \\ {\widetilde{\mathbf{x}}} _3 \end{array}\right] + \left[ \begin{array}{c} {\mathbf{B}} _1- {\mathbf{P}} _1 \\ {\mathbf{B}} _2 \\ 0 \end{array}\right] {\mathbf{u}} ,\\ {\mathbf{y}}&= \left[ \begin{array}{ccc} ( {\mathbf{B}} _1+ {\mathbf{P}} _1)^T&{\mathbf{B}} _2^T&0\end{array}\right] \left[ \begin{array}{c} {\widetilde{\mathbf{x}}} _1 \\ {\widetilde{\mathbf{x}}} _2 \\ {\widetilde{\mathbf{x}}} _3 \end{array}\right] +( {\mathbf{S}} + {\mathbf{N}} ) {\mathbf{u}} , \nonumber \end{aligned}$$
(3)

where \(\lambda {\mathbf{E}} _1-( {\mathbf{J}} _1- {\mathbf{R}} _1)\) is regular and \( {\mathbf{B}} _2\) has full row rank. Also, the subsystem

$$\begin{aligned} \left[ \begin{array}{cc} {\mathbf{E}} _1 &{} 0 \\ 0 &{} 0 \end{array}\right] \left[ \begin{array}{c} \dot{ {\widetilde{\mathbf{x}}} }_1 \\ \dot{ {\widetilde{\mathbf{x}}} }_2 \end{array}\right]= & {} \left[ \begin{array}{cc} {\mathbf{J}} _1- {\mathbf{R}} _1 &{} 0 \\ 0 &{} 0 \end{array}\right] \left[ \begin{array}{c} {\widetilde{\mathbf{x}}} _1 \\ {\widetilde{\mathbf{x}}} _2 \end{array}\right] + \left[ \begin{array}{c} {\mathbf{B}} _1- {\mathbf{P}} _1 \\ {\mathbf{B}} _2 \end{array}\right] {\mathbf{u}} ,\\ {\mathbf{y}}= & {} \left[ \begin{array}{cc} ( {\mathbf{B}} _1+ {\mathbf{P}} _1)^T&{\mathbf{B}} _2^T \end{array}\right] \left[ \begin{array}{c} {\widetilde{\mathbf{x}}} _1 \\ {\widetilde{\mathbf{x}}} _2 \end{array}\right] +( {\mathbf{S}} + {\mathbf{N}} ) {\mathbf{u}} , \nonumber \end{aligned}$$
(4)

obtained by removing the third equation and the variable \( {\mathbf{x}} _3\), is still a pHDAE.

Proof

First determine an orthogonal matrix \( {\mathbf{V}} _1\) such that

$$ {\mathbf{V}} _1^T (\lambda {\mathbf{E}} -( {\mathbf{J}} - {\mathbf{R}} )) {\mathbf{V}} _1= \lambda \left[ \begin{array}{cc} {\mathbf{E}} _1 &{} 0 \\ 0 &{} 0 \end{array}\right] - \left[ \begin{array}{cc} {\mathbf{J}} _1- {\mathbf{R}} _1 &{} 0 \\ 0 &{} 0 \end{array}\right] ,\ {\mathbf{V}} _1^T( {\mathbf{B}} - {\mathbf{P}} )= \left[ \begin{array}{c} {\mathbf{B}} _1- {\mathbf{P}} _1 \\ \tilde{\mathbf{B}} _2-\tilde{\mathbf{P}} _2 \end{array}\right] . $$

Such \( {\mathbf{V}} _1\) exists since \( {\mathbf{E}} , {\mathbf{J}} , {\mathbf{R}} \) have a common nullspace when the pencil \(\lambda {\mathbf{E}} -( {\mathbf{J}} - {\mathbf{R}} )\) is singular [23]. Then a row compression of \(\tilde{\mathbf{B}} _2-\tilde{\mathbf{P}} _2\) via an orthogonal matrix \(\tilde{\mathbf{V}} _2\) and a congruence transformation with \( {\mathbf{V}} _2=\textsf {diag}( {\mathbf{I}} , \tilde{ {\mathbf{V}} }_2)\) is performed, so that with \( {\mathbf{V}} = \textsf {diag}( {\mathbf{V}} _1, {\mathbf{V}} _2)\), we obtain the zero pattern in (3). Updating the output equation accordingly and using the fact that the transformed passivity matrix

$$ \tilde{ {\mathbf{W}} }=\begin{bmatrix} {\mathbf{V}} ^T {\mathbf{R}} {\mathbf{V}} &{} {\mathbf{V}} ^T {\mathbf{P}} \\ {\mathbf{P}} ^T {\mathbf{V}} &{} {\mathbf{S}} \end{bmatrix}\in \mathbb {R}^{(n+m)\times (n+m)} $$

is still semidefinite; it follows that \( {\mathbf{P}} _2=0\) and \( {\mathbf{P}} _3=0\), giving the desired form. \(\square \)

The next result presents a condensed form, showing that conditions (C2) and (O2) are equivalent and hold for the system in (4).

Lemma 2

For the pHDAE in (4) there exists an orthogonal basis transformation \( \widehat{ {\mathbf{V}} }\) in the state space and \( {\mathbf{U}} \) in the control space such that in the new variables \( \widehat{ {\mathbf{x}} }= \left[ \begin{array}{ccccc} \widehat{ {\mathbf{x}} }_1^{\,T}&\widehat{ {\mathbf{x}} }_2^{\,T}&\widehat{ {\mathbf{x}} }_3^{\,T} \widehat{ {\mathbf{x}} }_4^{\,T}&\widehat{ {\mathbf{x}} }_5^{\,T}&\widehat{ {\mathbf{x}} }_6^{\,T} \end{array}\right] ^T = \widehat{ {\mathbf{V}} }^T \left[ \begin{array}{cc} {\widetilde{\mathbf{x}}} _1^{\,T}&{\widetilde{\mathbf{x}}} _2^{\,T} \end{array}\right] ^T\), and \(\left[ \begin{array}{ccc } {\mathbf{u}} _1^T&{\mathbf{u}} _2^T&{\mathbf{u}} _3^T \end{array}\right] ^T= {\mathbf{U}} ^T {\mathbf{u}} \) the system has the form

(5)

where \( {\mathbf{E}} _{22}\), \( {\mathbf{J}} _{33}- {\mathbf{R}} _{33}\), \( {\mathbf{J}} _{15}\), and \( {\mathbf{B}} _{42}\) and \( {\mathbf{B}} _{63}\) are invertible. Furthermore, the subsystem obtained by deleting the first, fifth, and sixth block row and column satisfies (C2) and equivalently (O2).

Proof

The proof follows again by a sequence of orthogonal transformations. Starting from (4) in the first step one determines an orthogonal matrix \( {\mathbf{V}} _1\) (via a spectral decomposition of \( {\mathbf{E}} _1\)) such \( {\widetilde{\mathbf{V}}} _1^T {\mathbf{E}} _1 {\widetilde{\mathbf{V}}} _1= \left[ \begin{array}{cc} {\mathbf{E}} _{11} &{} 0 \\ 0 &{} 0 \end{array}\right] \) with \( {\mathbf{E}} _{11}>0\), and then forms a congruence transformation with \( \widehat{ {\mathbf{V}} }_1=\textsf {diag}( {\widetilde{\mathbf{V}}} _1, {\mathbf{I}} )\) yielding

$$ \widehat{ {\mathbf{V}} }_1^T( {\mathbf{J}} - {\mathbf{R}} ) \widehat{ {\mathbf{V}} }_1=\left[ \begin{array}{cc} \tilde{\mathbf{J}} _{11}-\tilde{\mathbf{R}} _{11} &{} \tilde{\mathbf{J}} _{12}-\tilde{\mathbf{R}} _{12} \\ \tilde{\mathbf{J}} _{21}-\tilde{\mathbf{R}} _{21}&{} \tilde{\mathbf{J}} _{22}-\tilde{\mathbf{R}} _{22} \end{array}\right] ,\ \widehat{ {\mathbf{V}} }_1^T( {\mathbf{B}} - {\mathbf{P}} )= \left[ \begin{array}{c} \tilde{\mathbf{B}} _1-\tilde{\mathbf{P}} _1 \\ \tilde{\mathbf{B}} _2-\tilde{\mathbf{P}} _2\end{array}\right] . $$

Next compute a full rank decomposition \( {\widetilde{\mathbf{V}}} _2^T (\widetilde{\mathbf{J}} _{22}-\widetilde{\mathbf{R}} _{22}) {\widetilde{\mathbf{V}}} _2= \left[ \begin{array}{cc} \widehat{\mathbf{J}} _{22} -\widehat{\mathbf{R}} _{22} &{} 0 \\ 0&{} 0 \end{array}\right] \), where \(\widehat{\mathbf{J}} _{22} -\widehat{\mathbf{R}} _{22}\) is invertible and \(\widehat{\mathbf{R}} _{22}\ge 0\). This exists, since \(\widetilde{\mathbf{J}} _{22}-\widetilde{\mathbf{R}} _{22}\) has a negative semidefinite symmetric part. Then an appropriate congruence transformation with \( \widehat{ {\mathbf{V}} }_2=\textsf {diag}( {\mathbf{I}} , {\widetilde{\mathbf{V}}} _2, {\mathbf{I}} )\) yields

$$\begin{aligned} \widehat{ {\mathbf{V}} }^T_2 \widehat{ {\mathbf{V}} }_1^T {\mathbf{E}} \widehat{ {\mathbf{V}} }_1 \widehat{ {\mathbf{V}} }_2= & {} \left[ \begin{array}{cccc} {\mathbf{E}} _{11} &{} 0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{}0 &{}0\\ 0 &{} 0 &{}0 &{}0 \end{array}\right] ,\\ \widehat{ {\mathbf{V}} }^T_2 \widehat{ {\mathbf{V}} }_1^T( {\mathbf{J}} - {\mathbf{R}} ) \widehat{ {\mathbf{V}} }_1 \widehat{ {\mathbf{V}} }_2= & {} \left[ \begin{array}{cccc} \widehat{\mathbf{J}} _{11}-\widehat{\mathbf{R}} _{11} &{} \widehat{\mathbf{J}} _{12}-\widehat{\mathbf{R}} _{12} &{} \widehat{\mathbf{J}} _{13} &{} 0\\ \widehat{\mathbf{J}} _{21}-\widehat{\mathbf{R}} _{21} &{} \widehat{\mathbf{J}} _{22}-\widehat{\mathbf{R}} _{22} &{}0 &{}0 \\ \widehat{\mathbf{J}} _{31} &{} 0&{} 0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0\end{array}\right] ,\ \widehat{ {\mathbf{V}} }_2^T \widehat{ {\mathbf{V}} }_1^T( {\mathbf{B}} - {\mathbf{P}} )= \left[ \begin{array}{c} \widehat{\mathbf{B}} _1-\widehat{\mathbf{P}} _1 \\ \widehat{\mathbf{B}} _2-\widehat{\mathbf{P}} _2 \\ \widehat{\mathbf{B}} _3 \\ \widehat{\mathbf{B}} _4\end{array}\right] , \end{aligned}$$

where \(\widehat{\mathbf{J}} _{22}-\widehat{\mathbf{R}} _{22}\) is invertible and \(\widehat{\mathbf{B}} _4\) has full row rank.

Then one performs an orthogonal decomposition

$$ {\widetilde{\mathbf{V}}} _3^T \left[ \begin{array}{c} \widehat{\mathbf{B}} _3 \\ \widehat{\mathbf{B}} _4 \end{array}\right] {\mathbf{U}} = \left[ \begin{array}{ccc} 0 &{} {\mathbf{B}} _{42} &{} {\mathbf{B}} _{43}\\ 0 &{} 0 &{} {\mathbf{B}} _{53}\\ 0 &{} 0 &{} {\mathbf{B}} _{63} \end{array}\right] $$

with \( {\mathbf{B}} _{42}\) and \( {\mathbf{B}} _{63}\) square nonsingular, where the number of rows in \( {\mathbf{B}} _{63}\) is that of \(\widehat{\mathbf{B}} _4\) and applies an appropriate congruence transformation with \( \widehat{ {\mathbf{V}} }_3=\textsf {diag}( {\mathbf{I}} , {\mathbf{I}} , \widehat{ {\mathbf{V}} }_3 , {\mathbf{I}} )\) so that one obtains block matrices

$$ \left[ \begin{array}{ccccc} {\mathbf{E}} _{1} &{} 0 &{} 0 &{} 0 &{}0\\ 0 &{} 0 &{} 0&{}0&{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{}0 \\ 0 &{} 0 &{} 0 &{}0 &{} 0\\ 0 &{} 0 &{} 0 &{}0 &{} 0\end{array}\right] , \left[ \begin{array}{ccccc} \bar{\mathbf{J}} _{11}-\bar{\mathbf{R}} _{11} &{} \bar{\mathbf{J}} _{12}-\bar{\mathbf{R}} _{12}&{} \bar{\mathbf{J}} _{13}-\bar{\mathbf{R}} _{13}&{}\bar{\mathbf{J}} _{14} &{}0\\ \bar{\mathbf{J}} _{21}-\bar{\mathbf{R}} _{21} &{} \bar{\mathbf{J}} _{22}-\bar{\mathbf{R}} _{22}&{} 0 &{} 0&{}0\\ \bar{\mathbf{J}} _{31} &{} 0&{} 0 &{} 0 &{} 0\\ \bar{\mathbf{J}} _{41}&{}0 &{} 0 &{} 0 &{}0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0\end{array}\right] , $$

As a final step one computes a column compression of the full row rank matrix \(\bar{\mathbf{J}} _{41}\) and applies an appropriate congruence transformation. This yields the desired form.

The fact that the several \( {\mathbf{P}} \) blocks do not occur follows again from the semidefiniteness of the transformed passivity matrix. Since \( {\mathbf{J}} _{51}\), \( {\mathbf{B}} _{42}\) and \( {\mathbf{B}} _{63}\) are invertible, it follows immediately that \( {\mathbf{u}} _3=0\) and \(\hat{\mathbf{x}} _1=0\) and that \(\hat{\mathbf{x}} _5\) is uniquely determined by all the other variables. Considering the subsystem obtained by removing the first, fifth, and sixth block row and column, it follows from the symmetry structure that the condition (C2) holds if and only (O2) holds.

The procedure to compute the condensed form (3) immediately separates the dynamical part (given by the first block row), the algebraic index-1 conditions (with and without dissipation, given by the second and third block rows), and the index-2 conditions (given by the fourth block row). The last row is the singular part of the free system. However, since the conditions (C2) and (O2) hold, the system can be made regular by output feedback. Note that this is already displayed in the subsystem (4), which, for ease of notation, we denote by

$$ \begin{array}{rl} {\mathbf{E}} _r\dot{ {\mathbf{x}} }_r\!\!\!\!&{}=( {\mathbf{J}} _r- {\mathbf{R}} _r)\, {\mathbf{x}} _r+( {\mathbf{B}} _r- {\mathbf{P}} _r)\, {\mathbf{u}} ,\\[2mm] {\mathbf{y}} _r\!\!\!\!&{}=( {\mathbf{B}} _r+ {\mathbf{P}} _r)^T\, {\mathbf{x}} _r+( {\mathbf{S}} _r+ {\mathbf{N}} _r)\, {\mathbf{u}} , \end{array} \qquad {\mathbf{x}} _r(t_0)=0. $$

If we apply an output feedback \( {\mathbf{u}} =- {\mathbf{K}} _r {\mathbf{y}} _r\) with \( {\mathbf{K}} _r= {\mathbf{K}} _r^T> 0 \) that makes the (2,2) block in the closed loop system invertible, then this corresponds to a feedback for the state-to-output map in (4) of the form \( {\mathbf{y}} _r=( {\mathbf{I}} +( {\mathbf{S}} _r+ {\mathbf{N}} _r) {\mathbf{K}} _r)^{-1}( {\mathbf{B}} _r+ {\mathbf{P}} _r)^T {\mathbf{x}} _r\), which we can insert into the first equation to obtain

$$\begin{aligned} {\mathbf{E}} _r\dot{ {\mathbf{x}} _r}= & {} ( {\mathbf{J}} _r- {\mathbf{R}} _r) {\mathbf{x}} _r+( {\mathbf{B}} _r- {\mathbf{P}} _r) {\mathbf{K}} _r {\mathbf{y}} _r\\= & {} \left( ( {\mathbf{J}} _r- {\mathbf{R}} _r)-( {\mathbf{B}} _r- {\mathbf{P}} _r) ( {\mathbf{K}} _r^{-1} + {\mathbf{S}} _r+ {\mathbf{N}} _r)^{-1}( {\mathbf{B}} _r+ {\mathbf{P}} _r)^T\right) {\mathbf{x}} _r. \end{aligned}$$

The system matrix is the Schur complement of

$$ \left[ \begin{array}{rr} \ {\mathbf{J}} _r &{} {\mathbf{B}} _r \\[2mm] \ - {\mathbf{B}} _r^T &{} \quad - {\mathbf{N}} _r\end{array}\right] - \left[ \begin{array}{cc} \ {\mathbf{R}} _r &{} {\mathbf{P}} _r \\[2mm] \ {\mathbf{P}} _r^T &{} \quad {\mathbf{K}} _r^{-1}+ {\mathbf{S}} _r\end{array}\right] , $$

which has a symmetric part given by \(-\widetilde{\mathbf{W}} _r\), with \( \widetilde{\mathbf{W}} _r=\left[ \begin{array}{cc} \ {\mathbf{R}} _r &{} {\mathbf{P}} _r \\[2mm] \ {\mathbf{P}} _r^T &{} \quad {\mathbf{K}} _r^{-1}+ {\mathbf{S}} _r\end{array}\right] \ge 0. \) Hence, the closed loop system is regular and the closed loop system is still pH.

The procedures described above are computationally demanding, since they typically require large-scale singular value decompositions or spectral decompositions. Fortunately, in many practical cases the condensed form is already available directly from the modeling procedure, so that the transfer function can be formed and the model reduction method can be directly applied.

4 Interpolatory Model Reduction of pHDAEs

Given an order-n pHDAE as in (1), we want to construct an order-r reduced pHDAE, with \(r \ll n\), having the same structured form

$$\begin{aligned} \begin{aligned} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r&=( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} })\, {\mathbf{x}} _r+( \widehat{ {\mathbf{B}} }- \widehat{ {\mathbf{P}} })\, {\mathbf{u}} ,\quad {\mathbf{x}} _r(t_0)=0,\\ {\mathbf{y}} _r&=( \widehat{ {\mathbf{B}} }+ \widehat{ {\mathbf{P}} })^T\, {\mathbf{x}} _r+( \widehat{ {\mathbf{S}} }+ \widehat{ {\mathbf{N}} })\, {\mathbf{u}} , \end{aligned} \end{aligned}$$
(6)

such that \( \widehat{ {\mathbf{E}} }\), \( \widehat{ {\mathbf{J}} }\), \( \widehat{ {\mathbf{R}} }\in {}\mathsf {I}\!\mathsf {R}^{r \times r}\), \( \widehat{ {\mathbf{B}} }, \widehat{ {\mathbf{P}} }\in {}\mathsf {I}\!\mathsf {R}^{r \times m}\), \( \widehat{ {\mathbf{S}} }= \widehat{ {\mathbf{S}} }^T\), \( \widehat{ {\mathbf{N}} }=- \widehat{ {\mathbf{N}} }^T\in {}\mathsf {I}\!\mathsf {R}^{m \times m}\) satisfy the same requirements as in Definition 1 and that the output \( {\mathbf{y}} _r(t)\) of (6) is an accurate approximation to the original output \( {\mathbf{y}} (t)\) over a wide range of admissible inputs \( {\mathbf{u}} (t)\). We will enforce accuracy by constructing the reduced model (6) via interpolation.

Let \( {\mathbf{H}} (s) = {\boldsymbol{\mathcal {C}}} (s {\mathbf{E}} - {\mathbf{A}} )^{-1} {\boldsymbol{\mathcal {B}}} + {\mathbf{D}} \) and \( \widehat{ {\mathbf{H}} }(s) = \boldsymbol{\widehat{\mathcal {C}}} (s \widehat{ {\mathbf{E}} }- \widehat{ {\mathbf{A}} })^{-1} \boldsymbol{\widehat{\mathcal {B}}}+ \widehat{ {\mathbf{D}} }\) denote the transfer functions of (1) and (6), where \({\boldsymbol{\mathcal {C}}} = ( {\mathbf{B}} + {\mathbf{P}} )^T\), \( {\mathbf{A}} = {\mathbf{J}} - {\mathbf{R}} \), \({\boldsymbol{\mathcal {B}}} = {\mathbf{B}} - {\mathbf{P}} \), \( {\mathbf{D}} = {\mathbf{S}} + {\mathbf{N}} \), and similarly for the reduced-order (“hat”) quantities. Given right-interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\} \in {}\mathsf {I}\!\mathsf {C}\) together with the corresponding right-tangent directions \(\{\boldsymbol{b}_1,\boldsymbol{b}_2,\ldots ,\boldsymbol{b}_r\} \in {}\mathsf {I}\!\mathsf {C}^m\) and left-interpolation points \(\{\mu _1,\mu _2,\ldots ,\mu _r\} \in {}\mathsf {I}\!\mathsf {C}\) together with the corresponding left-tangent directions \(\{\boldsymbol{\ell }_1,\boldsymbol{\ell }_2,\ldots ,\boldsymbol{\ell }_r\} \in {}\mathsf {I}\!\mathsf {C}^m\), we wish to construct a reduced model, \( \widehat{ {\mathbf{H}} }(s)\), that tangentially interpolates \( {\mathbf{H}} (s)\), i.e.,

$$ {\mathbf{H}} (\sigma _i) \boldsymbol{b}_i = \widehat{ {\mathbf{H}} }(\sigma _i) \boldsymbol{b}_i~~~\text{ and }~~~ \boldsymbol{\ell }_i^T {\mathbf{H}} (\mu _i) = \boldsymbol{\ell }_i^T \widehat{ {\mathbf{H}} }(\mu _i),~~\text{ for }~~i=1,2,\ldots ,r.$$

These tangential interpolation conditions can be enforced easily via a Petrov-Galerkin projection [1, 4, 12]. In particular, construct \( {\mathbf{V}} \in {}\mathsf {I}\!\mathsf {C}^{n \times r} \) and \( {\mathbf{Z}} \in {}\mathsf {I}\!\mathsf {C}^{n \times r}\) using

$$\begin{aligned} {\mathbf{V}}&= \left[ (\sigma _1 {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_1,~~(\sigma _2 {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_2,~~\cdots ~(\sigma _r {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_r\right] ~~\text{ and }~~\\ {\mathbf{Z}}&= \left[ (\mu _1 {\mathbf{E}} - {\mathbf{A}} )^{-T}{\boldsymbol{\mathcal {C}}} ^T \boldsymbol{\ell }_1,~~(\mu _2 {\mathbf{E}} - {\mathbf{A}} )^{-T}{\boldsymbol{\mathcal {C}}} ^T \boldsymbol{\ell }_2,~~\cdots ~(\mu _r {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {C}}} ^T \boldsymbol{\ell }_r\right] . \end{aligned}$$

Then an interpolatory reduced model can be defined via the projection:

$$\begin{aligned} \widehat{ {\mathbf{E}} }= {\mathbf{Z}} ^T {\mathbf{E}} {\mathbf{V}} ,~~ \widehat{ {\mathbf{A}} }= {\mathbf{Z}} ^T {\mathbf{A}} {\mathbf{V}} ,~~\boldsymbol{\widehat{\mathcal {B}}}= {\mathbf{Z}} ^T {\boldsymbol{\mathcal {B}}} ,~~\boldsymbol{\widehat{\mathcal {C}}} ={\boldsymbol{\mathcal {C}}} {\mathbf{V}} ,~~\text{ and }~~ \widehat{ {\mathbf{D}} }= {\mathbf{D}} . \end{aligned}$$
(7)

In the setting of pHDAEs, two fundamental issues arise: First, the reduced quantities in (7) are no longer guaranteed to have pH structure. This is most easily seen by examining the reduced quantity, \(\boldsymbol{\widehat{\mathcal {A}}} = {\mathbf{Z}} ^T {\boldsymbol{\mathcal {A}}} {\mathbf{V}} = {\mathbf{Z}} ^T {\mathbf{J}} {\mathbf{V}} - {\mathbf{Z}} ^T {\mathbf{R}} {\mathbf{V}} \). If \(\boldsymbol{\widehat{\mathcal {A}}} \) is decomposed into its symmetric and skew-symmetric parts, it can no longer be guaranteed that the symmetric part is positive semi-definite. This could be resolved by using a Ritz-Galerkin projection, taking \( {\mathbf{Z}} = {\mathbf{V}} \), but then only interpolation conditions associated with right interpolation data are satisfied. Even then, there will still be a second issue that persists. In the generic case when \(r < \mathop {\mathrm {rank}}\nolimits ( {\mathbf{E}} )\), the reduced quantity \( \widehat{ {\mathbf{E}} }\) will generally be nonsingular; thus the reduced system will be an ODE and the polynomial parts of \( {\mathbf{H}} (s)\) and \( \widehat{ {\mathbf{H}} }(s)\) will not match, leading then to unbounded errors.

Structure-preserving interpolatory reduction of pH systems in the most general setting of tangential interpolation has been studied in [15, 16]. However, this work focused on the ODE case. On the other hand, [17] developed the tangential interpolation framework for reducing unstructured DAEs with guaranteed polynomial matching. Only recently in [11, 18], the combined problem has been investigated. We now develop a treatment of structure-preserving interpolatory model reduction problem for index-1 and index-2 pHDAEs in the general setting of tangential interpolation.

4.1 Semi-explicit Index-1 pHDAE Systems

The simplest class of pHDAEs are semi-explicit index-1 pHDAEs of the form

$$\begin{aligned} \begin{array}{rcl} \left[ \begin{array} {cc} {\mathbf{E}} _{11} &{} 0 \\ 0 &{} 0 \end{array} \right] \dot{ {\mathbf{x}} }(t)&{} = &{} \left[ \begin{array} {rr} {\mathbf{J}} _{11}- {\mathbf{R}} _{11} &{} {\mathbf{J}} _{12}- {\mathbf{R}} _{12} \\ - {\mathbf{J}} _{12}^T- {\mathbf{R}} _{12}^T &{} {\mathbf{J}} _{22}- {\mathbf{R}} _{22} \end{array} \right] {\mathbf{x}} (t) + \left[ \begin{array} {c} {\mathbf{B}} _{1}- {\mathbf{P}} _1 \\ {\mathbf{B}} _2- {\mathbf{P}} _2 \end{array} \right] u(t), \\[3ex] {\mathbf{y}} (t) &{}= &{} \left[ \begin{array} {cc} {\mathbf{B}} _1^T+ {\mathbf{P}} _1^T&{} {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T \end{array} \right] {\mathbf{x}} (t) + ( {\mathbf{S}} + {\mathbf{N}} ) {\mathbf{u}} (t), \end{array} \end{aligned}$$
(8)

where \( {\mathbf{E}} _{11}\) and \( {\mathbf{J}} _{22}- {\mathbf{R}} _{22}\) are nonsingular. We have the following interpolation result.

Theorem 1

Consider the pHDAE system in (8). Let the interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\} \in {}\mathsf {I}\!\mathsf {C}\) and the corresponding tangent directions \(\{\boldsymbol{b}_1,\boldsymbol{b}_2,\ldots ,\boldsymbol{b}_r\} \in {}\mathsf {I}\!\mathsf {C}^m\) be given. Construct the interpolatory model reduction basis \( {\mathbf{V}} \) as

$$\begin{aligned} {\mathbf{V}} = \begin{bmatrix} {\mathbf{V}} _1 \\ {\mathbf{V}} _2 \end{bmatrix} = \left[ (\sigma _1 {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_1, ~~\cdots ,~~(\sigma _r {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_r\right] \in {}\mathsf {I}\!\mathsf {C}^{n \times r}, \end{aligned}$$
(9)

where \( {\mathbf{V}} \) is partitioned conformably with the system, and define the matrices

$$\begin{aligned} {\mathbb B} = \begin{bmatrix}\boldsymbol{b}_1&\boldsymbol{b}_2&\cdots&\boldsymbol{b}_r \end{bmatrix}\in {}\mathsf {I}\!\mathsf {C}^{m \times r} ~ \text{ and } ~ {\boldsymbol{\mathcal {D}}} = {\mathbf{D}} - ( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) ( {\mathbf{J}} _{22}- {\mathbf{R}} _{22})^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) \in {}\mathsf {I}\!\mathsf {C}^{m \times m}. \end{aligned}$$

Let \( {\mathbf{A}} =\left[ \begin{array}{cc} {\mathbf{A}} _{11} &{} {\mathbf{A}} _{12}\\ {\mathbf{A}} _{21} &{} {\mathbf{A}} _{22} \end{array}\right] = {\mathbf{J}} - {\mathbf{R}} , \) partitioned accordingly to (8). Then, the transfer function \( \widehat{ {\mathbf{H}} }(s)\) of the reduced model

$$\begin{aligned} \begin{array}{rclcrcl} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r(t)= & {} ( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }) {\mathbf{x}} _r(t) + \boldsymbol{\widehat{\mathcal {B}}} {\mathbf{u}} (t),&~~~~&{\mathbf{y}} _r(t)= & {} \boldsymbol{\widehat{\mathcal {C}}} {\mathbf{x}} _r(t) + \widehat{ {\mathbf{D}} } {\mathbf{u}} (t), \end{array} \end{aligned}$$
(10)

with

$$\begin{aligned} \begin{array}{rclcrcl} \widehat{ {\mathbf{E}} }&{}= &{} {\mathbf{V}} _1^T {\mathbf{E}} _{11} {\mathbf{V}} _1, &{} ~~~ &{} \boldsymbol{\widehat{\mathcal {C}}} &{}=&{} {\boldsymbol{\mathcal {C}}} {\mathbf{V}} + ( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) {\mathbf{A}} _{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) {\mathbb B} , \\ \widehat{ {\mathbf{A}} }&{}= &{} {\mathbf{V}} ^T {\boldsymbol{\mathcal {A}}} {\mathbf{V}} + {\mathbb B} ^T ({\boldsymbol{\mathcal {D}}} - {\mathbf{D}} ) {\mathbb B} , &{} ~~~ &{} \widehat{ {\mathbf{D}} }&{}=&{} {\boldsymbol{\mathcal {D}}} = {\mathbf{D}} - ( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) {\mathbf{A}} _{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2),\nonumber \\ \boldsymbol{\widehat{\mathcal {B}}}&{}=&{} {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} + {\mathbb B} ^T( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) {\mathbf{A}} _{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2), \end{array} \end{aligned}$$

matches the polynomial part of \( {\mathbf{H}} (s)\) and tangentially interpolates it, i.e.,

$$ {\mathbf{H}} (\sigma _i) \boldsymbol{b}_i = \widehat{ {\mathbf{H}} }(\sigma _i) \boldsymbol{b}_i,~~\text{ for }~~i=1,2,\ldots ,r. $$

Define \( \widehat{ {\mathbf{P}} }=\frac{1}{2} (-\boldsymbol{\widehat{\mathcal {G}}} + \boldsymbol{\widehat{\mathcal {C}}} )\), and decompose \( \widehat{ {\mathbf{D}} }= \widehat{ {\mathbf{S}} }+ \widehat{ {\mathbf{N}} }\) and \( \widehat{ {\mathbf{A}} }= \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }\) into their symmetric and skew-symmetric parts. Then, the reduced model (10) is a pHDAE system if the reduced passivity matrix \( \widehat{\mathbf{W}} =\left[ \begin{array}{cc} \widehat{ {\mathbf{R}} }&{} \widehat{ {\mathbf{P}} }\\ \widehat{ {\mathbf{P}} }^T &{} \widehat{ {\mathbf{S}} }\end{array}\right] \) is positive seimdefinite.

Proof

We employ a Galerkin projection using the interpolatory model reduction basis \( {\mathbf{V}} \) to obtain an intermediate reduced model

$$\begin{aligned} \widetilde{ {\mathbf{E}} }= {\mathbf{V}} _1^T {\mathbf{E}} _{11} {\mathbf{V}} _1,~~ \widetilde{ {\mathbf{A}} }= \widetilde{ {\mathbf{J}} }- \widetilde{ {\mathbf{R}} }= {\mathbf{V}} ^T {\mathbf{J}} {\mathbf{V}} - {\mathbf{V}} ^T {\mathbf{R}} {\mathbf{V}} ,~~ \boldsymbol{\widetilde{\mathcal {B}}} = {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} , ~~\boldsymbol{\widetilde{\mathcal {C}}}= {\boldsymbol{\mathcal {C}}} {\mathbf{V}} ,~~\text{ and }~~ \widetilde{ {\mathbf{D}} }= {\mathbf{D}} . \end{aligned}$$

This reduced model is a pHDAE system due to the use of a one-sided Galerkin projection, and likewise it also satisfies the desired tangential interpolation conditions. However, it will not generally match the transfer function, \( {\mathbf{H}} (s)\), at \(s=\infty \). Indeed, \( {\mathbf{H}} (s)\) has a polynomial part is given by

$$ \lim _{s\rightarrow \infty } {\mathbf{H}} (s) = {\boldsymbol{\mathcal {D}}} = {\mathbf{D}} - ( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T) {\mathbf{A}} _{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2)\ne \widetilde{ {\mathbf{D}} }= \lim _{s\rightarrow \infty } \widehat{ {\mathbf{H}} }(s) $$

A remedy to this problem, proposed in [2, 21] and employed in the general DAE setting in [17], is to modify the \( {\mathbf{D}} \)-term in the reduced model to match the polynomial part while at the same time shifting other reduced quantities appropriately so as to retain tangential interpolation. Using this approach, we obtain a modified reduced model

$$\begin{aligned} \widehat{ {\mathbf{E}} }&= \widetilde{ {\mathbf{E}} }= {\mathbf{V}} _1^T {\mathbf{E}} _{11} {\mathbf{V}} _1,~~ \nonumber \\ \widehat{ {\mathbf{A}} }&= \widetilde{ {\mathbf{A}} }+ {\mathbb B} ^T ({\boldsymbol{\mathcal {D}}} - {\mathbf{D}} ) {\mathbb B} = {\mathbf{V}} ^T {\mathbf{J}} {\mathbf{V}} - {\mathbf{V}} ^T {\mathbf{R}} {\mathbf{V}} - {\mathbb B} ^T( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) {\mathbb B} ,\nonumber \\ \boldsymbol{\widehat{\mathcal {B}}}&= \boldsymbol{\widetilde{\mathcal {B}}} + {\mathbb B} ^T( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) \nonumber \\&= {\mathbf{V}} _1^T ( {\mathbf{B}} _1 - {\mathbf{P}} _1)+ {\mathbf{V}} _2^T ( {\mathbf{B}} _2 - {\mathbf{P}} _2)+ {\mathbb B} ^T( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2),\nonumber \\ \boldsymbol{\widehat{\mathcal {C}}}&= \boldsymbol{\widetilde{\mathcal {C}}}+ ( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) {\mathbb B} \nonumber \\&= ( {\mathbf{B}} _1^T+ {\mathbf{P}} _1^T) {\mathbf{V}} _1^T + ( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T) {\mathbf{V}} _2^T +( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2) {\mathbb B} ,~~~~\text{ and }\nonumber \\ \widehat{ {\mathbf{D}} }&= {\boldsymbol{\mathcal {D}}} = {\mathbf{D}} - ( {\mathbf{B}} _2^T + {\mathbf{P}} _2^T)A_{22}^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2),\nonumber \end{aligned}$$

which satisfies the original tangential interpolation conditions while also matching the polynomial part with a modified \( \widehat{ {\mathbf{D}} }\)-term. For this system to be pH, we need to check that the associated passivity matrix is still positive semidefinite. After rewriting the input and output matrix in the usual way, this is exactly the condition on \(\widehat{\mathbf{W}} \) in the assertion. We then have that the reduced model not only satisfies the interpolation conditions and matches the polynomial part at \(s=\infty \), but also is pH. \(\square \)

Remark 1

Note that if the input does not influence the algebraic equations, i.e., if \( {\mathbf{B}} _2 = {\mathbf{P}} _2=0\), then the shift of the constant term is not necessary and the formulas simplify significantly, i.e., \( \widehat{ {\mathbf{A}} }= {\mathbf{V}} ^T {\boldsymbol{\mathcal {A}}} {\mathbf{V}} \), \(\boldsymbol{\widehat{\mathcal {B}}}= \boldsymbol{\widetilde{\mathcal {B}}} = {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} \), \(\boldsymbol{\widehat{\mathcal {C}}} = {\boldsymbol{\mathcal {C}}} {\mathbf{V}} \), and \( \widehat{ {\mathbf{D}} }= {\boldsymbol{\mathcal {D}}} = {\mathbf{D}} \).

Another solution to preserving pH structure via interpolation can be obtained through the following theorem.

Theorem 2

Consider a full-order pHDAE system of the form (8). Let interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\} \in {}\mathsf {I}\!\mathsf {C}\) and the corresponding tangent directions \(\{\boldsymbol{b}_1,\boldsymbol{b}_2,\ldots ,\boldsymbol{b}_r\} \in {}\mathsf {I}\!\mathsf {C}^m\) be given. Construct the interpolatory model reduction basis \( {\mathbf{V}} \) as in (9). Then the reduced model

$$\begin{aligned} \begin{array}{rcl} \left[ \begin{array} {cc} {\mathbf{V}} _1^T {\mathbf{E}} _{11} {\mathbf{V}} _1 &{} 0 \\ 0 &{} 0 \end{array} \right] \dot{ {\mathbf{x}} }(t)&{} = &{} \left[ \begin{array} {rr} { {\mathbf{V}} }_1^T( {\mathbf{J}} _{11}- {\mathbf{R}} _{11}){ {\mathbf{V}} }_1 &{} { {\mathbf{V}} }_1^T( {\mathbf{J}} _{12}- {\mathbf{R}} _{12}) \\ (- {\mathbf{J}} _{12}^T- {\mathbf{R}} _{12}^T){ {\mathbf{V}} }_1 &{} {\mathbf{J}} _{22}- {\mathbf{R}} _{22} \end{array} \right] {\mathbf{x}} (t) + \left[ \begin{array} {c} { {\mathbf{V}} }_1^T( {\mathbf{B}} _{1}- {\mathbf{P}} _1) \\ {\mathbf{B}} _2- {\mathbf{P}} _2 \end{array} \right] {\mathbf{u}} (t) \\[3ex] {\mathbf{y}} _r(t) &{}= &{} \left[ \begin{array} {cc} ( {\mathbf{B}} _1^T+ {\mathbf{P}} _1)^T{ {\mathbf{V}} }_1&{} {\mathbf{B}} _2+ {\mathbf{P}} _2^T \end{array} \right] {\mathbf{x}} (t) + {\mathbf{D}} {\mathbf{u}} (t), \end{array} ~ \end{aligned}$$
(11)

retains the pH structure, tangentially interpolates the original model, and matches the polynomial part.

Proof

We first note that the subspace spanned by the columns of \( {\mathbf{V}} = \begin{bmatrix} {\mathbf{V}} _1^T&{\mathbf{V}} _2^T \end{bmatrix}^T \) is contained in the subspace spanned by the columns of \( \widehat{ {\mathbf{V}} } := \textsf {diag}( {\mathbf{V}} _1, {\mathbf{I}} ). \) Then, the system in (11) results from reducing the original system in (8) via \(\widehat{ {\mathbf{V}} }\). Since \(\mathsf {span}( {\mathbf{V}} ) \subseteq \mathsf {span}(\widehat{ {\mathbf{V}} })\), this reduced DAE automatically satisfies the interpolation conditions and since \( \widehat{ {\mathbf{V}} }\) does not alter the matrix \( {\mathbf{J}} _{22} - {\mathbf{R}} _{22}\) and the matrices \( {\mathbf{B}} _2, {\mathbf{P}} _2\), the polynomial part of the transfer function is \( {\mathbf{D}} - ( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) ( {\mathbf{J}} _{22} - {\mathbf{R}} _{22})^{-1} ( {\mathbf{B}} _2- {\mathbf{P}} _2)\), matching that of the original model. The reduced system in (11) is pH as well since the one-sided projection that is used retains the original pH structure. \(\square \)

Remark 2

Theorem 2 appears to present an easier alternative to the more complicated Theorem 1 for structure-preserving interpolatory model reduction of index-1 pHDAEs. However, the construction of Theorem 2 may not achieve the maximal reduction possible because redundant algebraic conditions cannot be removed; see [25]. Note also that further orthogonalization of \( {\mathbf{V}} _1\) may be necessary, see [11].

4.2 Semi-explicit pHDAE Systems with Index-2 Constraints

The next class of interest will be semi-explicit index-2 systems. We first consider the case that the input does not affect the algebraic equations.

Theorem 3

Consider an index-2 pHDAE system of the form

$$\begin{aligned} \begin{array}{rcl} \left[ \begin{array} {cc} {\mathbf{E}} _{11} &{} 0 \\ 0 &{} 0 \end{array} \right] \left[ \begin{array}{c} \dot{ {\mathbf{x}} }_1(t) \\ \dot{ {\mathbf{x}} }_2(t) \end{array} \right] &{} = &{} \left[ \begin{array} {cc} {\mathbf{J}} _{11}- {\mathbf{R}} _{11} &{} {\mathbf{J}} _{12} \\ - {\mathbf{J}} _{12}^T &{} \mathbf {0} \end{array} \right] \left[ \begin{array}{c} { {\mathbf{x}} }_1(t) \\ { {\mathbf{x}} }_2(t) \end{array} \right] + \left[ \begin{array} {c} {\mathbf{B}} _{1} - {\mathbf{P}} _1 \\ \mathbf {0} \end{array} \right] {\mathbf{u}} (t), \\[3ex] {\mathbf{y}} (t) &{}= &{} \left[ \begin{array} {cc} {\mathbf{B}} _1^T + {\mathbf{P}} _1^T&{} \mathbf {0} \end{array} \right] \left[ \begin{array}{c} { {\mathbf{x}} }_1(t) \\ { {\mathbf{x}} }_2(t) \end{array} \right] + {\mathbf{D}} {\mathbf{u}} (t), \end{array} ~ \end{aligned}$$
(12)

with \( {\mathbf{E}} _{11}>0\) and set \( {\mathbf{A}} _{11}= {\mathbf{J}} _{11}- {\mathbf{R}} _{11}\). Given interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\}\) and associated tangent directions \(\{ \boldsymbol{b}_1, \boldsymbol{b}_2,\dots , \boldsymbol{b}_r\}\), let the vectors \( {\mathbf{v}} _i\), for \(i=1,2,\ldots ,r\), be the first block of the solution of

$$\begin{aligned} \left[ \begin{array}{cc} {\mathbf{A}} _{11}-\sigma _i {\mathbf{E}} _{11} &{} {\mathbf{J}} _{12}\\ - {\mathbf{J}} _{12}^T &{} \mathbf {0} \end{array} \right] \left[ \begin{array}{cc} {\mathbf{v}} _i \\ {\mathbf{z}} \end{array} \right] = \left[ \begin{array}{cc} ( {\mathbf{B}} _1- {\mathbf{P}} _1)\boldsymbol{b}_i \\ \mathbf {0} \end{array} \right] . \end{aligned}$$
(13)

Define \( {\mathbf{V}} = \left[ {\mathbf{v}} _1, {\mathbf{v}} _2, \dots , {\mathbf{v}} _r\right] \). Then, the reduced model

$$\begin{aligned} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r = ( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }) {\mathbf{x}} _r + \boldsymbol{\widehat{\mathcal {B}}} {\mathbf{u}} (t),~~ {\mathbf{y}} _r = \boldsymbol{\widehat{\mathcal {C}}} {\mathbf{x}} _r + \widehat{ {\mathbf{D}} }, \end{aligned}$$
(14)
$$\begin{aligned} \text{ with } \widehat{ {\mathbf{E}} }&= {\mathbf{V}} ^T {\mathbf{E}} _{11} {\mathbf{V}} ,~~ \nonumber \widehat{ {\mathbf{J}} }= {\mathbf{V}} ^T {\mathbf{J}} _{11} {\mathbf{V}} ,~~ \widehat{ {\mathbf{R}} }= {\mathbf{V}} ^T {\mathbf{R}} _{11} {\mathbf{V}} , \\ \boldsymbol{\widehat{\mathcal {B}}}&= {\mathbf{V}} ^T {\mathbf{B}} _1 - {\mathbf{V}} ^T {\mathbf{P}} _1,~~\boldsymbol{\widehat{\mathcal {C}}} = {\mathbf{B}} _1^T {\mathbf{V}} ^T + {\mathbf{P}} ^T {\mathbf{V}} _1^T,~~~~\text{ and }~~~ \widehat{ {\mathbf{D}} }= {\mathbf{D}} , \end{aligned}$$
(15)

is still pH, matches the polynomial part of the original transfer function, and satisfies the tangential interpolation conditions,

$$ {\mathbf{H}} (\sigma _i) \boldsymbol{b}_i = \widehat{ {\mathbf{H}} }(\sigma _i) \boldsymbol{b}_i,~~\text{ for }~~i=1,2,\ldots ,r. $$

Proof

Note first that the regularity of \(\lambda {\mathbf{E}} -( {\mathbf{J}} - {\mathbf{R}} )\) and the index-2 condition imply that \(- {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12}\) is invertible, see [5, 20]. Following [17], we write (12) as

$$\begin{aligned} \begin{array}{rcl} {\boldsymbol{\Pi }} {\mathbf{E}} _{11}{\boldsymbol{\Pi }}^T \dot{ {\mathbf{x}} }_1(t) &{} = &{} {\boldsymbol{\Pi }} {\mathbf{A}} _{11}{\boldsymbol{\Pi }}^T {\mathbf{x}} _1(t) + {\boldsymbol{\Pi }} {\mathbf{B}} _1 {\mathbf{u}} (t), \\ {\mathbf{y}} (t) &{}=&{} {\mathbf{C}} _1{\boldsymbol{\Pi }}^T {\mathbf{x}} _1(t) + {\mathbf{D}} {\mathbf{u}} (t), \end{array} \end{aligned}$$
(16)

and conjoin this with the algebraic equation,

$$ {\mathbf{x}} _2(t)= -( {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12})^{-1} {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{A}} _{11} {\mathbf{x}} _1(t) -( {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12})^{-1} {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{B}} _1 {\mathbf{u}} (t). $$

The skew projector, \({\boldsymbol{\Pi }}\), in (16) is defined as \({\boldsymbol{\Pi }}= {\mathbf{I}} - {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12}( {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12})^{-1} {\mathbf{J}} _{12}^T, \) and (16) is now an implicit ODE pH system that can be reduced with standard model reduction techniques. As it stands, this would appear to require computing the projector \({\boldsymbol{\Pi }}\) explicitly, see [25]. For general index-2 DAE systems one can avoid this computational step through interpolatory model reduction, see [17].

To adapt this idea to pHDAE systems, we construct \( {\mathbf{V}} \) using (13) and then compute the reduced-order quantities via one-sided projection as in (15). This construction of \( {\mathbf{V}} \), as in [17], guarantees that the reduced model in (14) tangentially interpolates the original pHDAE system in (16) and the polynomial part of the transfer function in (12) is given by \( {\mathbf{D}} = {\mathbf{S}} + {\mathbf{N}} \), separated with respect to its symmetric and skew-symmetric parts. Since (14) is an implicit ODE pH system with an exact \( {\mathbf{D}} \)-term, it matches the polynomial part of the original transfer function \( {\mathbf{H}} (s)\).

It remains to show that (14) is pH. By construction in (15), \( \widehat{ {\mathbf{J}} }\) is skew-symmetric, \( \widehat{ {\mathbf{R}} }\) is symmetric positive semidefinite, and \( {\mathbf{E}} _{11}\) is symmetric positive definite. Moreover,

$$ \left[ \begin{array}{cc} {\mathbf{V}} ^T {\mathbf{R}} _{11} {\mathbf{V}} &{} {\mathbf{V}} ^T {\mathbf{P}} _1 \\ {\mathbf{P}} _1^T {\mathbf{V}} &{} {\mathbf{S}} \end{array} \right] = \left[ \begin{array}{cc} {\mathbf{V}} ^T &{} 0 \\ 0 &{} {\mathbf{I}} \end{array} \right] \left[ \begin{array}{cc} {\mathbf{R}} _{11} &{} {\mathbf{P}} _1 \\ {\mathbf{P}} _1^T &{} {\mathbf{S}} \end{array} \right] \left[ \begin{array}{cc} {\mathbf{V}} &{} 0 \\ 0 &{} {\mathbf{I}} \end{array} \right] \ge 0, $$

since the original model is pH. Therefore, the pH-structure is retained. \(\square \)

The situation becomes more complicated when the second block in \({\boldsymbol{\mathcal {B}}} \) is nonzero, that is, when the system has the form

$$\begin{aligned} \begin{array}{rcl} \left[ \begin{array} {cc} {\mathbf{E}} _{11} &{} 0 \\ 0 &{} 0 \end{array} \right] \left[ \begin{array}{c} \dot{ {\mathbf{x}} }_1(t) \\ \dot{ {\mathbf{x}} }_2(t) \end{array} \right] &{} =&{} \left[ \begin{array} {cc} {\mathbf{J}} _{11}- {\mathbf{R}} _{11} &{} {\mathbf{J}} _{12} \\ - {\mathbf{J}} _{12}^T &{} \mathbf {0} \end{array} \right] \left[ \begin{array}{c} { {\mathbf{x}} }_1(t) \\ { {\mathbf{x}} }_2(t) \end{array} \right] + \left[ \begin{array} {c} {\mathbf{B}} _{1}- {\mathbf{P}} _1 \\ {\mathbf{B}} _2- {\mathbf{P}} _2 \end{array} \right] {\mathbf{u}} (t), \\[3ex] {\mathbf{y}} (t) &{}= &{} \left[ \begin{array} {cc} {\mathbf{B}} _1^T+ {\mathbf{P}} _1^T&{} {\mathbf{B}} _2+ {\mathbf{P}} _2^T \end{array} \right] \left[ \begin{array}{c} { {\mathbf{x}} }_1(t) \\ { {\mathbf{x}} }_2(t) \end{array} \right] + {\mathbf{D}} {\mathbf{u}} (t). \end{array} ~ \end{aligned}$$
(17)

Theorem 4

Consider a pHDAE system of the form (17) and define the matrices

$$ \begin{array}{c} {\mathcal Z} = -( {\mathbf{A}} _{21} {\mathbf{E}} _{11}^{-1} {\mathbf{A}} _{12})^{-1} = \left( {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12}\right) ^{-1}, \\[2mm] {\boldsymbol{\mathcal {B}}} = ( {\mathbf{B}} _1 - {\mathbf{P}} _1) + ( {\mathbf{J}} _{11}- {\mathbf{R}} _{11}) {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12} {\mathbf{Z}} ( {\mathbf{B}} _2- {\mathbf{P}} _2), \\[2mm] {\boldsymbol{\mathcal {C}}} = ( {\mathbf{B}} _1^T- {\mathbf{P}} _1^T) - ( {\mathbf{B}} _2^T- {\mathbf{P}} _2^T)\, {\mathcal Z} \, {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1}( {\mathbf{J}} _{11}- {\mathbf{R}} _{11}),\\[3mm] \begin{array}{cc} {\boldsymbol{\mathcal {D}}} _0 = {\mathbf{D}} - ( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T)\, {\mathcal Z} \, {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1}( {\mathbf{B}} _1- {\mathbf{P}} _1),&{}\ \text{ and } \ {\boldsymbol{\mathcal {D}}} _1 = -( {\mathbf{B}} _2^T+ {\mathbf{P}} _2^T) \, {\mathcal Z} \, ( {\mathbf{B}} _2- {\mathbf{P}} _2). \end{array} \end{array} $$

Given interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\}\) and associated tangent directions \(\{ \boldsymbol{b}_1, \boldsymbol{b}_2,\dots , \boldsymbol{b}_r\}\), let the vectors \( {\mathbf{v}} _i\) be the first blocks of the solutions of

$$\begin{aligned} \left[ \begin{array}{cc} {\mathbf{J}} _{11}- {\mathbf{R}} _{11}-\sigma _i {\mathbf{E}} _{11} &{} {\mathbf{J}} _{12}\\ - {\mathbf{J}} _{12}^T &{} \mathbf {0} \end{array} \right] \left[ \begin{array}{cc} {\mathbf{v}} _i \\ {\mathbf{z}} \end{array} \right] = {\mathbf{B}} \boldsymbol{b}_i,~~\text{ for }~~i=1,2,\ldots ,r, \end{aligned}$$
(18)

and set \( {\mathbf{V}} := \left[ {\mathbf{v}} _1, {\mathbf{v}} _2, \dots , {\mathbf{v}} _r\right] \), \( {\mathbf{u}} _1:= {\mathbf{u}} \), \( {\mathbf{u}} _2:=\dot{ {\mathbf{u}} }\), and \( \widehat{ {\mathbf{D}} }:=\left[ \begin{array}{cc} {\boldsymbol{\mathcal {D}}} _0&{\boldsymbol{\mathcal {D}}} _1 \end{array}\right] = \widehat{ {\mathbf{S}} }+ \widehat{ {\mathbf{N}} }\). Then, the reduced model

$$\begin{aligned} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r = ( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }) {\mathbf{x}} _r + \left[ \begin{array}{cc} \widehat{ {\mathbf{B}} }- \widehat{ {\mathbf{P}} }&0\end{array}\right] \left[ \begin{array}{c} {\mathbf{u}} _1\\ {\mathbf{u}} _2\end{array}\right] ,~~ {\mathbf{y}} _r = ( \widehat{ {\mathbf{B}} }+ \widehat{ {\mathbf{P}} })^T {\mathbf{x}} _r + \widehat{ {\mathbf{D}} }\left[ \begin{array}{c} {\mathbf{u}} _1\\ {\mathbf{u}} _2\end{array}\right] , \end{aligned}$$
(19)
$$\begin{aligned} \text{ with } \widehat{ {\mathbf{E}} }&= {\mathbf{V}} ^T {\mathbf{E}} _{11} {\mathbf{V}} ,~~ \widehat{ {\mathbf{J}} }= {\mathbf{V}} ^T {\mathbf{J}} _{11} {\mathbf{V}} ,~~ \widehat{ {\mathbf{R}} }= {\mathbf{V}} ^T {\mathbf{R}} _{11} {\mathbf{V}} ,\nonumber \\ \widehat{ {\mathbf{B}} }&= \frac{1}{2}\left( {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} + {\mathbf{V}} ^T {\boldsymbol{\mathcal {C}}} ^T\right) ,~~\text{ and }~~ \widehat{ {\mathbf{P}} }= \frac{1}{2}\left( {\mathbf{V}} ^T {\boldsymbol{\mathcal {C}}} ^T- {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} \right) , \end{aligned}$$
(20)

satisfies the interpolation conditions, matches the polynomial part of the transfer function, and preserves the pH structure, provided that the reduced passivity matrix \(\widehat{\mathbf{W}} =\left[ \begin{array}{cc} \widehat{ {\mathbf{R}} }&{} \widehat{ {\mathbf{P}} }\\ \widehat{ {\mathbf{P}} }^T &{} \widehat{ {\mathbf{S}} }\end{array}\right] \) is positive semidefinite.

Proof

The proof is similarly to that of Theorem 3. Following [17], the state \( {\mathbf{x}} _1\) can be decomposed as \( {\mathbf{x}} _1 = {\mathbf{x}} _c + {\mathbf{x}} _g\), where \( {\mathbf{x}} _g = {\mathbf{E}} _{11}^{-1} {\mathbf{A}} _{12}( {\mathbf{J}} _{12}^T {\mathbf{E}} _{11}^{-1} {\mathbf{J}} _{12})^{-1}( {\mathbf{B}} _2- {\mathbf{P}} _2) {\mathbf{u}} (t)\) and \( {\mathbf{x}} _c(t)\) satisfies \( {\mathbf{J}} _{12}^T {\mathbf{x}} _c = 0\). Then, one can rewrite (17) as

$$\begin{aligned} \begin{array}{rcl} {\boldsymbol{\Pi }} {\mathbf{E}} _{11}{\boldsymbol{\Pi }}^T \dot{ {\mathbf{x}} }_c(t) &{} = &{} {\boldsymbol{\Pi }} {\mathbf{A}} _{11}{\boldsymbol{\Pi }}^T {\mathbf{x}} _c(t) + {\boldsymbol{\Pi }} {\mathbf{B}} _1 {\mathbf{u}} (t), \\ {\mathbf{y}} (t) &{}=&{} {\mathbf{C}} {\boldsymbol{\Pi }}^T {\mathbf{x}} _c(t) +{\boldsymbol{\mathcal {D}}} _0 {\mathbf{u}} (t) + {\boldsymbol{\mathcal {D}}} _1\dot{ {\mathbf{u}} }(t). \end{array} \end{aligned}$$
(21)

As before, the ODE part can be reduced with usual model reduction techniques. Following [17], however, we achieve this without computing the projector \({\boldsymbol{\Pi }}\) explicitly, instead by constructing \( {\mathbf{V}} \) using (18) and then applying one-sided model reduction with \( {\mathbf{V}} \) to obtain the reduced model

$$\begin{aligned} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r = ( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }) {\mathbf{x}} _r + \widetilde{ {\mathbf{B}} } {\mathbf{u}} (t),~~ {\mathbf{y}} _r = \widetilde{ {\mathbf{C}} }^T {\mathbf{x}} _r + {\boldsymbol{\mathcal {D}}} _0 {\mathbf{u}} (t) + {\boldsymbol{\mathcal {D}}} _1 \dot{ {\mathbf{u}} }(t), \end{aligned}$$
(22)

where \( \widehat{ {\mathbf{E}} }= {\mathbf{V}} ^T {\mathbf{E}} _{11} {\mathbf{V}} \), \( \widehat{ {\mathbf{J}} }= {\mathbf{V}} ^T {\mathbf{J}} _{11} {\mathbf{V}} \), \( \widehat{ {\mathbf{R}} }= {\mathbf{V}} ^T {\mathbf{R}} _{11} {\mathbf{V}} \), \( \widetilde{ {\mathbf{B}} }= {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} \), and \( \widetilde{ {\mathbf{C}} }= {\boldsymbol{\mathcal {C}}} {\mathbf{V}} \). This reduced model, by construction, satisfies the tangential interpolation conditions. Note that the reduced model in (22) has exactly the same realization as the reduced model in (19) except for the reduced \( \widetilde{ {\mathbf{B}} }\) and \( \widetilde{ {\mathbf{C}} }\) terms. The reduced terms \( \widehat{ {\mathbf{E}} }\), \( \widehat{ {\mathbf{J}} }\), and \( \widehat{ {\mathbf{R}} }\) in (22) already have the pH structure. To recover the port symmetry, we determine matrices \( \widehat{ {\mathbf{B}} }\) and \( \widehat{ {\mathbf{P}} }\) such that \( \widetilde{ {\mathbf{B}} }= {\mathbf{V}} ^T {\boldsymbol{\mathcal {B}}} = \widehat{ {\mathbf{B}} }- \widehat{ {\mathbf{P}} }\) and \( \widetilde{ {\mathbf{C}} }= {\boldsymbol{\mathcal {C}}} {\mathbf{V}} = ( \widehat{ {\mathbf{B}} }+ \widehat{ {\mathbf{P}} })^T\) via

$$ \widehat{ {\mathbf{B}} }= \frac{1}{2}\left( \widetilde{ {\mathbf{B}} }+ \widetilde{ {\mathbf{C}} }^T\right) = \frac{1}{2} {\mathbf{V}} ^T \left( {\boldsymbol{\mathcal {B}}} +{\boldsymbol{\mathcal {C}}} ^T\right) \quad \text{ and } \quad \widehat{ {\mathbf{P}} }= \frac{1}{2}\left( \widetilde{ {\mathbf{C}} }^T - \widetilde{ {\mathbf{B}} }\right) = \frac{1}{2} {\mathbf{V}} ^T \left( {\boldsymbol{\mathcal {C}}} ^T - {\boldsymbol{\mathcal {B}}} \right) ,$$

recovering (20). The final requirement to retain the pH structure is, then, again that \(\left[ \begin{array}{cc} \widehat{ {\mathbf{R}} }&{} \widehat{ {\mathbf{P}} }\\ \widehat{ {\mathbf{P}} }^T &{} \frac{1}{2}\left( {\mathbf{D}} + {\mathbf{D}} ^T \right) \end{array} \right] \ge 0\), which is the final condition in the statement of the theorem. \(\square \)

This approach has the disadvantage that \( {\mathbf{u}} _2 = \dot{ {\mathbf{u}} }\) is introduced as an extra input and this, in turn, may lead to difficulties when applying standard control and optimization methods. For this reason it is usually preferable to first perform index reduction via an appropriate output feedback (see Sect. 3) and then apply the results from Sect. 4.1. Note that this will change the polynomial part of the transfer function.

4.3 Semi-explicit pHDAE Systems with Index-1 and Index-2 Constraints

Finally, we consider semi-explicit index-2 systems that also have an index-1 part, see [11, 18, 25]. We will only consider the special case,

$$\begin{aligned} \left[ \begin{array}{ccc} {\mathbf{E}} _{11} &{} {\mathbf{E}} _{22} &{} 0 \\ {\mathbf{E}} _{21} &{} {\mathbf{E}} _{22} &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right] \left[ \begin{array}{c} \dot{ {\mathbf{x}} }_1 \\ \dot{ {\mathbf{x}} }_2 \\ \dot{ {\mathbf{x}} }_3 \end{array}\right] = \left[ \begin{array}{ccc} {\mathbf{J}} _{11}- {\mathbf{R}} _{11} &{}\ {\mathbf{J}} _{12}- {\mathbf{R}} _{12}&{}\ {\mathbf{J}} _{13} \\ {\mathbf{J}} _{21}- {\mathbf{R}} _{21} &{}\ {\mathbf{J}} _{22}- {\mathbf{R}} _{22} &{}\ 0 \\ {\mathbf{J}} _{31} &{}\ 0 &{}\ 0 \end{array}\right] \left[ \begin{array}{c} {\mathbf{x}} _1 \\ {\mathbf{x}} _2 \\ {\mathbf{x}} _3\end{array}\right] + \left[ \begin{array}{c} {\mathbf{B}} _1- {\mathbf{P}} _1 \\ {\mathbf{B}} _2- {\mathbf{P}} _2 \\ 0\end{array}\right] {\mathbf{u}} ,\\ {\mathbf{y}} = \left[ \begin{array}{ccc} ( {\mathbf{B}} _1+ {\mathbf{P}} _1)^T&( {\mathbf{B}} _2+ {\mathbf{P}} _2)^T&0\end{array}\right] \left[ \begin{array}{c} {\mathbf{x}} _1 \\ {\mathbf{x}} _2 \\ {\mathbf{x}} _3 \end{array}\right] +( {\mathbf{S}} + {\mathbf{N}} ) {\mathbf{u}} ,~\text{ where }~ \left[ \begin{array}{cc} {\mathbf{E}} _{11} &{} {\mathbf{E}} _{22} \\ {\mathbf{E}} _{21} &{} {\mathbf{E}} _{22}\end{array}\right] >0, \nonumber \end{aligned}$$
(23)

and both \( {\mathbf{J}} _{22} - {\mathbf{R}} _{22}\) and \( {\mathbf{J}} _{31}\) are nonsingular.

Theorem 5

Consider an index-2 pHDAE system of the form (23) and construct an interpolatory model reduction basis given by

$$\begin{aligned} {\mathbf{V}} = \begin{bmatrix} {\mathbf{V}} _1^T&{\mathbf{V}} _2^T&{\mathbf{V}} _3^T \end{bmatrix}^T = \left[ (\sigma _1 {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_1,~~\cdots ~(\sigma _r {\mathbf{E}} - {\mathbf{A}} )^{-1}{\boldsymbol{\mathcal {B}}} \boldsymbol{b}_r\right] \in {}\mathsf {I}\!\mathsf {C}^{n \times r}, \end{aligned}$$

partitioned to conform with the system. Define \(\widehat{ {\mathbf{V}} } := \textsf {diag}( {\mathbf{I}} , {\mathbf{V}} _2, {\mathbf{I}} )\).

Then, the reduced system

$$\begin{aligned} \widehat{ {\mathbf{E}} }\dot{ {\mathbf{x}} }_r = ( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }) {\mathbf{x}} _r + \boldsymbol{\widehat{\mathcal {B}}} {\mathbf{u}} (t),~~ {\mathbf{y}} _r = \boldsymbol{\widehat{\mathcal {C}}} {\mathbf{x}} _r + \widehat{ {\mathbf{D}} }, \end{aligned}$$
(24)
$$\begin{aligned} \text{ with }~~ \widehat{ {\mathbf{E}} }&= \widehat{\mathbf{V}} ^T {\mathbf{E}} \widehat{\mathbf{V}} ,~~ \nonumber \widehat{ {\mathbf{J}} }= \widehat{\mathbf{V}} ^T {\mathbf{J}} \widehat{\mathbf{V}} ,~~ \widehat{ {\mathbf{R}} }= \widehat{\mathbf{V}} ^T {\mathbf{R}} \widehat{\mathbf{V}} , \\ \boldsymbol{\widehat{\mathcal {B}}}&= \widehat{\mathbf{V}} ^T{\boldsymbol{\mathcal {B}}} = \widehat{\mathbf{V}} ^T {\mathbf{B}} - \widehat{\mathbf{V}} ^T {\mathbf{P}} ,~~\boldsymbol{\widehat{\mathcal {C}}} = {\mathbf{C}} \widehat{\mathbf{V}} = {\mathbf{B}} ^T \widehat{\mathbf{V}} ^T + {\mathbf{P}} ^T \widehat{\mathbf{V}} ^T,~~~~\text{ and }~~~ \widehat{ {\mathbf{D}} }= {\mathbf{D}} , \end{aligned}$$
(25)

is still pH, matches the polynomial part of the transfer function, and satisfies the tangential interpolation conditions, i.e., \( {\mathbf{H}} (\sigma _i) \boldsymbol{b}_i = \widehat{ {\mathbf{H}} }(\sigma _i) \boldsymbol{b}_i\), for \(i=1,2,\ldots ,r\).

Proof

It follows from the definitions of \( {\mathbf{V}} \) and \(\widehat{ {\mathbf{V}} }\) that \(\mathsf {span}( {\mathbf{V}} ) \subseteq \mathsf {span}(\widehat{ {\mathbf{V}} })\). Therefore, the resulting reduced system automatically satisfies the interpolation conditions. Since \( \widehat{ {\mathbf{V}} }\) does not alter the algebraic constraints, the polynomial part of its transfer function is still \( {\mathbf{D}} \), matching that of the original model. The reduced system in (24)-(25) is a pHDAE as this one-sided projection retains the original pH structure. \(\square \)

5 Algorithmic Considerations

The preceding analysis presumed that interpolation points and tangent directions were specified beforehand. We consider now how one might make choices that generally produce effective approximations with respect to the \( {{\mathcal H}_{2}} \) system measure.

5.1 \( {{\mathcal H}_{2}} \)-inspired Structure-Preserving Interpolation

The \( {{\mathcal H}_{2}} \) distance between the full model \( {\mathbf{H}} (s)\) and the reduced model \( {\mathbf{H}} _r(s)\) is

$$\begin{aligned} \big \Vert {\mathbf{H}} - {\mathbf{H}} _r\big \Vert _{ {{\mathcal H}_{2}} } = \left( \frac{1}{2\pi } \int _{-\infty }^\infty \big \Vert {\mathbf{H}} (\imath \omega ) - {\mathbf{H}} _r(\imath \omega )\big \Vert _F^2 d\omega \right) ^{1/2}, \end{aligned}$$

where \(\imath ^2 = -1\) and \(\Vert {\mathbf{M}} \Vert _F\) denote the Frobenius norm of a matrix \( {\mathbf{M}} \). To have a finite \( {{\mathcal H}_{2}} \) error norm the polynomial parts of \( {\mathbf{H}} _r(s)\) and \( {\mathbf{H}} (s)\) need to match. To make this precise, write \( {\mathbf{H}} (s)= {\mathbf{G}} (s) + {\mathbf{P}} (s)\) and \( {\mathbf{H}} _r(s)= {\mathbf{G}} _r(s) + {\mathbf{P}} _r(s)\) where \( {\mathbf{G}} (s)\) and \( {\mathbf{G}} _r(s)\) are strictly proper transfer functions, and \( {\mathbf{P}} (s)\) and \( {\mathbf{P}} _r(s)\) are the polynomial parts. Therefore, if \( {\mathbf{H}} _r(s)\) is the \( {{\mathcal H}_{2}} \)-optimal approximation to \( {\mathbf{H}} (s)\), then \( {\mathbf{P}} _r(s) = {\mathbf{P}} (s)\) and \( {\mathbf{G}} _r(s)\) is the \( {{\mathcal H}_{2}} \)-optimal approximation to \( {\mathbf{G}} (s)\). This suggests that one decompose \( {\mathbf{H}} (s)\) into its rational and polynomial parts \( {\mathbf{H}} (s) = {\mathbf{G}} (s) + {\mathbf{P}} (s)\) and then apply \( {{\mathcal H}_{2}} \) optimal reduction to \( {\mathbf{G}} (s)\). However, this requires the explicit construction of \( {\mathbf{G}} (s)\). This problem was resolved in [17] for unstructured index-1 and index-2 DAEs. On the other hand, for the ODE case, [16] proposed a pH structure preserving algorithm for minimizing the \( {{\mathcal H}_{2}} \) norm. In this section, we aim to unify these two approaches.

First, we briefly revisit the interpolatory \( {{\mathcal H}_{2}} \) optimality conditions; for details we refer the reader to [1, 4, 14] and the references therein. Since \( {{\mathcal H}_{2}} \) optimality for the DAE case boils down to optimality for the ODE part, we focus on the latter. Let \( {\mathbf{G}} _r(s) = \sum _{i=1}^r \frac{ {\mathbf{c}} _i {\mathbf{b}} _i^T}{s-\lambda _i}\) be the pole-residue decomposition of \( {\mathbf{G}} _r(s)\). For simplicity we assume simple poles. If \( {\mathbf{G}} _r(s)\) is an \( {{\mathcal H}_{2}} \) optimal approximation to \( {\mathbf{G}} (s)\), then \( {\mathbf{G}} (-\lambda _i) {\mathbf{b}} _i = {\mathbf{G}} _r(-\lambda _i) {\mathbf{b}} _i\), \( {\mathbf{c}} _i^T {\mathbf{G}} (-\lambda _i) = {\mathbf{c}} _i^T {\mathbf{G}} _r(-\lambda _i)\), and \( {\mathbf{c}} _i^T {\mathbf{G}} '(-\lambda _i) {\mathbf{b}} _i = {\mathbf{c}} _i^T {\mathbf{G}} '_r(-\lambda _i) {\mathbf{b}} _i,\) for \(i=1,2,\ldots ,r\) where \('\) denotes the derivative with respect to s. Therefore an \( {{\mathcal H}_{2}} \) optimal reduced model is a bitangential Hermite interpolant where the interpolation points are the mirror images of its poles and the tangent directions are the residue directions. Since the optimality conditions depend on the reduced model to be computed, this requires an iterative algorithm, such as the Iterative Rational Krylov Algorithm [14]. For other approaches to \( {{\mathcal H}_{2}} \) optimal approximation, see, e.g., [1, 4].

Following [16], to preserve the structure, we will satisfy only a subset of these conditions. We will enforce interpolation points to be the mirror images of the reduced order poles and enforce either left or right-tangential interpolation conditions (\( {\mathbf{Z}} = {\mathbf{V}} \)) without the derivative conditions. However, intuitively, one might expect that in the pH setting this may not cause too much deviation from true optimality, since the input-to-state matrix \( {\mathbf{B}} \) and the state-to-output matrix \( {\mathbf{C}} \) are related.

For simplicity, consider the semi-explicit index-2 case. The other scenarios follow similarly. Starting with the interpolation points \(\{\sigma _1,\sigma _2,\ldots ,\sigma _r\}\) and the tangent directions \(\{ \boldsymbol{b}_1, \boldsymbol{b}_2,\dots , \boldsymbol{b}_r\}\), construct the interpolatory reduced pHDAE \( \widehat{ {\mathbf{H}} }(s)\) as in Theorem 3. Let \( \widehat{ {\mathbf{H}} }(s) = \sum _{i=1}^r \frac{\widehat{\boldsymbol{\ell }}_i \widehat{\boldsymbol{b}}_i^T}{s-\lambda _i} + {\mathbf{D}} \) be the pole-residue decomposition. Since initially the optimality condition \(\sigma _i = -\lambda _i( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }, \widehat{ {\mathbf{E}} })\) is not (generally) satisfied, choose \(-\lambda _i( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }, \widehat{ {\mathbf{E}} })\) as the next set of interpolation points and \(\widehat{\boldsymbol{b}}_i\) as the next set of tangent directions. This is repeated until convergence upon which the reduced model \( {\mathbf{H}} _r(s)\) is not only a structure-preserving pHDAE, but also satisfies \(\sigma _i = -\lambda _i( \widehat{ {\mathbf{J}} }- \widehat{ {\mathbf{R}} }, \widehat{ {\mathbf{E}} })\).

5.2 Numerical Example

We illustrate the discussed procedure with the incompressible fluid flow model of the Oseen equations, from [18, §4.1]:

$$\begin{aligned} \begin{array}{rclcrclr} \partial _t \mathsf {v} &{} = &{} -(\mathbf {a} \cdot \nabla )\mathsf {v} + \mu \Delta \mathsf {v} - \nabla p + f &{} ~\text{ in } ~\Omega \times (0,T],~\quad ~ &{}\mathsf {v} &{} = &{}0, &{} \text{ on }~ \partial \Omega \times (0,T], \\ 0 &{} = &{} -\text{ div } \, \mathsf {v}, &{} ~\text{ in }~ \Omega \times (0,T], ~\quad ~&{}\mathsf {v} &{} = &{}\mathsf {v}^0, &{} \text{ in }~ \Omega \times {0}, \end{array} \end{aligned}$$

where \(\mathsf {v}\) and p are the velocity and pressure variables, \(\mu =1\) is the dynamic viscosity, \(\mathbf {a} = [\,1\ 1\,]^T\) is the convective velocity, and \(\Omega = (0,1)^2\) with boundary \(\partial \Omega \). f is an externally imposed body force that, for simplicity, is assumed to be separable: \(f(x,t)=b(x)u(t)\). A finite-difference discretization on a staggered rectangular grid leads to a single-input/single-output index-2 pHDAE of the form (12), see [18]. In our model, we used a uniform grid with 50 grid points yielding a descriptor system pHDAE of order \(n=7399\), of which \(n_1=4900\) degrees of freedom are for velocity and \(n_2=2499\) for pressure. We initialize our \(\mathcal {H}_2\)-based approach with logarithmically spaced interpolation points in the interval \([10^{-2},10^4]\) and construct a reduced model of the form (14) with orders \(r=1,2,\ldots ,10\). For every r value, we compute \(\Vert {\mathbf{H}} - \widehat{ {\mathbf{H}} }\Vert _\infty / \Vert {\mathbf{H}} \Vert _\infty \) where \(\Vert {\mathbf{H}} \Vert _\infty = \sup _{\omega \in {}\mathsf {I}\!\mathsf {R}} \mid {\mathbf{H}} (\imath \omega )\mid \). Figure 1 shows that the reduced transfer function accurately approximates the original one; for \(r=10\), the relative error is around \(10^{-5}\).

Fig. 1
figure 1

Evolution of the model reduction error for Oseen example as r varies

Full size image

6 Conclusions

We have presented interpolatory model reduction methods for several classes of large scale linear time-invariant port-Hamiltonian differential-algebraic systems. We have shown how constraints can be represented in the transfer function in such a way that the polynomial part can be preserved with interpolatory methods while still retaining important system structure. The results were illustrated with a numerical example from flow control.