Port-Hamiltonian Descriptor Systems

• Port-Hamiltonian descriptor systems are differential-algebraic frameworks that merge energy storage, interconnection, and dissipation into one unified representation.
• They leverage algebraic constraints to maintain power balance and passivity, facilitating structure-preserving stability analysis and feedback design.
• They find applications in electrical circuits, fluid networks, and PDEs, with numerical techniques emphasizing preservation of their intrinsic energy properties.

Port-Hamiltonian descriptor systems are differential-algebraic systems that combine the energy-based modeling paradigm of port-Hamiltonian systems with the algebraic constraints of descriptor systems. In linear time-invariant form they are typically written as

$$E\dot x=(J-R)Qx+(B-P)u,\qquad y=(B+P)^{T}Qx+(S+N)u,$$

or, in a simplified variant, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$, with structure matrices satisfying skew-symmetry, semidefinite dissipation, and the compatibility condition $Q^{T}E=E^{T}Q\ge 0$. The framework encodes interconnection, energy storage, dissipation, and ports in a single descriptor representation, and it is designed to retain power-balance and passivity under constraints, transformations, and interconnections (Beattie et al., 2017, Gernandt et al., 2021, Mehrmann et al., 2022).

1. Algebraic definition and power balance

A linear time-invariant descriptor system has the form

$E\dot x = Ax+Bu,\qquad y=Cx+Du,$

with $(E,A)$ regular if $\det(\lambda E-A)\not\equiv 0$ for at least one $\lambda\in\mathbb C$ (Chu et al., 2024, Cherifi et al., 2022). In the port-Hamiltonian descriptor setting, the coefficients are factorized as

$$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$

subject to

$$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$

with $J=-J^T$, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$0, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$1, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$2 (Cherifi et al., 2022, Chu et al., 2024). A commonly used homogeneous specialization is

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$3

and when $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$4 this is called a dissipative-Hamiltonian descriptor system (Gernandt et al., 2021).

The Hamiltonian is the quadratic stored-energy functional

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$5

or, in the complex notation used in some sources,

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$6

and $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$7 ensures nonnegativity (Chu et al., 2 Sep 2025, Cherifi et al., 2022). Along sufficiently regular trajectories, the power balance reads

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$8

while in the simplified form it reduces to

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$9

or, with $Q^{T}E=E^{T}Q\ge 0$0,

$Q^{T}E=E^{T}Q\ge 0$1

(Chu et al., 2 Sep 2025, Chu et al., 2024, Beattie et al., 2017). This identity is the central reason pH descriptor systems are naturally passive and Lyapunov stable in the zero-input case (Mehrmann et al., 2019, Mehrmann et al., 2022).

The descriptor matrix $Q^{T}E=E^{T}Q\ge 0$2 may be singular. This is not an accessory feature but part of the modeling framework: pHDAEs extend port-Hamiltonian ODEs to constrained systems and to high-index descriptor formulations while preserving the energetic interpretation (Beattie et al., 2017, Mehrmann et al., 2022).

2. Geometric structure, invariance, and equivalent representations

The descriptor formulation has a geometric counterpart in terms of Dirac, Lagrange, and maximal resistive structures. In the geometric picture, a trajectory is specified by storage variables, efforts, resistive variables, and external ports satisfying

$Q^{T}E=E^{T}Q\ge 0$3

and the Dirac orthogonality together with the resistive inequality yields

$Q^{T}E=E^{T}Q\ge 0$4

(Gernandt et al., 2023). A one-to-one correspondence between this geometric formulation and the descriptor realization by pHDAEs is established by explicit constructions in both directions (Gernandt et al., 2023).

The pHDAE class is invariant under congruence and basis changes. For time-varying invertible $Q^{T}E=E^{T}Q\ge 0$5 and $Q^{T}E=E^{T}Q\ge 0$6, the transformed coefficients

$Q^{T}E=E^{T}Q\ge 0$7

again define a pHDAE, and the transformed Hamiltonian satisfies $Q^{T}E=E^{T}Q\ge 0$8 (Beattie et al., 2017). The same invariance principle appears in nonlinear pHDAEs under diffeomorphic coordinate changes, where $Q^{T}E=E^{T}Q\ge 0$9 and the transformed system preserves the pH form (Mehrmann et al., 2019).

Power-conserving interconnection is equally intrinsic. Two subsystems

$E\dot x = Ax+Bu,\qquad y=Cx+Du,$0

can be interconnected by static power-preserving relations such as $E\dot x = Ax+Bu,\qquad y=Cx+Du,$1, and the aggregate system remains port-Hamiltonian with total Hamiltonian $E\dot x = Ax+Bu,\qquad y=Cx+Du,$2 (Mehrmann et al., 2022, Mehrmann et al., 2019). This invariance under interconnection is one reason the framework is described as ideal for automated network-based modeling (Mehrmann et al., 2022).

The same structural ideas extend beyond finite-dimensional ODE/DAE settings. For PDEs, Schöberl and Siuka formulate infinite-dimensional pH systems directly from the power-balance relation, allowing the Hamiltonian density $E\dot x = Ax+Bu,\qquad y=Cx+Du,$3 to depend on derivative variables. In the non-differential-operator case one recovers

$E\dot x = Ax+Bu,\qquad y=Cx+Du,$4

while in the differential-operator case one writes

$E\dot x = Ax+Bu,\qquad y=Cx+Du,$5

and the energy balance acquires boundary-port terms from integration by parts (Schöberl et al., 2012). For 1D distributed systems, explicit descriptor formulations and implicit Stokes-Lagrange subspace representations are linked by bijective transformations that commute with flow-constraint projections (Bendimerad-Hohl et al., 2024).

3. Regularity, index, system space, and stability

Regularity and differentiation index are basic analytical notions for pH descriptor systems. The pencil $E\dot x = Ax+Bu,\qquad y=Cx+Du,$6 is regular if $E\dot x = Ax+Bu,\qquad y=Cx+Du,$7 for some $E\dot x = Ax+Bu,\qquad y=Cx+Du,$8; equivalently it has exactly $E\dot x = Ax+Bu,\qquad y=Cx+Du,$9 finite eigenvalues (Chu et al., 2 Sep 2025). In Weierstraß or strict Kronecker form, the nilpotent blocks at infinity determine the differentiation index. In port-Hamiltonian DAEs one always has $(E,A)$0, but index two is troublesome because solutions may not exist for arbitrary inputs and impulses or non-smooth phenomena may appear (Chu et al., 2 Sep 2025, Chu et al., 2024).

The system space

$(E,A)$1

is the natural invariant space on which descriptor trajectories evolve (Gernandt et al., 2021). In quasi-Kronecker coordinates, it is obtained by deleting components associated with inconsistent or purely algebraic blocks (Gernandt et al., 2021). This viewpoint is essential in stability theory because a singular descriptor pair need not define dynamics on the whole ambient space.

A generalized Lyapunov inequality characterizes behavioral stability. For a homogeneous DAE $(E,A)$2, stability is equivalent to regularity of $(E,A)$3, spectral inclusion $(E,A)$4 with semisimple imaginary-axis eigenvalues, and the existence of a symmetric matrix $(E,A)$5 satisfying

$(E,A)$6

on the system space $(E,A)$7 (Gernandt et al., 2021). If such an $(E,A)$8 exists, then on $(E,A)$9 the DAE can be rewritten as a dissipative-Hamiltonian descriptor system via

$\det(\lambda E-A)\not\equiv 0$0

with $\det(\lambda E-A)\not\equiv 0$1, $\det(\lambda E-A)\not\equiv 0$2, and $\det(\lambda E-A)\not\equiv 0$3 on $\det(\lambda E-A)\not\equiv 0$4 (Gernandt et al., 2021). This gives a converse-to-structure statement: every behaviorally stable DAE admits a dH representation on its system space.

For homogeneous pH DAEs $\det(\lambda E-A)\not\equiv 0$5, sufficient and necessary conditions for stability are available under $\det(\lambda E-A)\not\equiv 0$6. In that case, when $\det(\lambda E-A)\not\equiv 0$7 is invertible,

$\det(\lambda E-A)\not\equiv 0$8

(Gernandt et al., 2021). A related geometric criterion states that, for dH systems with $\det(\lambda E-A)\not\equiv 0$9, regularity and stability are implied by

$\lambda\in\mathbb C$0

under the same nondegeneracy assumption (Gernandt et al., 2021).

A common misconception is that passivity alone implies asymptotic stability. The 2024 state-feedback results explicitly distinguish these notions: pH descriptor systems are known to be stable and passive, but they may not be asymptotically stable or strictly passive (Chu et al., 2024). This distinction drives much of the later feedback theory.

4. Regularization and stabilization by feedback

A central control problem for pH descriptor systems is to make a possibly non-regular or index-two closed loop regular, index at most one, asymptotically stable, and still port-Hamiltonian. For output feedback, the general proportional-derivative law is

$\lambda\in\mathbb C$1

which leads to the closed-loop system

$\lambda\in\mathbb C$2

(Chu et al., 2 Sep 2025). Three special cases are pure proportional feedback, pure derivative feedback, and mixed feedback (Chu et al., 2 Sep 2025).

For proportional feedback, there exists $\lambda\in\mathbb C$3 such that $\lambda\in\mathbb C$4 is regular, $\lambda\in\mathbb C$5, and the loop remains pH if and only if

$\lambda\in\mathbb C$6

where $\lambda\in\mathbb C$7 is any basis of the right nullspace of $\lambda\in\mathbb C$8 (Chu et al., 2 Sep 2025). For derivative feedback, there exists $\lambda\in\mathbb C$9 such that $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$0 is regular, $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$1, $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$2 is maximal among matrices of the form $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$3, and the closed loop is pH if and only if

$$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$4

equivalently $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$5 (Chu et al., 2 Sep 2025). Under complete observability and additional rank data, mixed proportional-derivative feedback allows prescribed rank $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$6 together with regularity, index reduction, and pH preservation (Chu et al., 2 Sep 2025).

The 2024 stabilization results treat the standard form

$$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$7

obtained after a standard reformulation so that $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$8 and $$\begin{pmatrix}A & B\ C & D\end{pmatrix} = \begin{pmatrix} (J-R)Q & G-P\ (G+P)^TQ & S+N \end{pmatrix},$$9 (Chu et al., 2024). With static output feedback

$$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$0

the closed-loop matrices are

$$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$1

where $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$2 and $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$3 with $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$4, $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$5 (Chu et al., 2024). Necessary and sufficient conditions are given for regularization, index reduction to $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$6, and asymptotic stabilization. In particular, proportional feedback achieves regularity and index $$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$7 exactly when

$$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$8

and asymptotic stabilization further requires

$$Q^TE=E^TQ\ge 0,\qquad \Gamma= \begin{pmatrix} J & G\ -G^T & N \end{pmatrix} =-\Gamma^T, \qquad W= \begin{pmatrix} Q^TRQ & Q^TP\ P^TQ & S \end{pmatrix}\ge 0,$$9

(Chu et al., 2024). Combined proportional and derivative feedback can enforce $J=-J^T$0, $J=-J^T$1, regularity, index $J=-J^T$2, and asymptotic stability under the corresponding rank conditions (Chu et al., 2024).

State-feedback results provide the analogous structure-preserving picture. For

$J=-J^T$3

the proportional state feedback $J=-J^T$4 yields

$J=-J^T$5

so the closed loop remains pH (Chu et al., 2024). There exists $J=-J^T$6 such that $J=-J^T$7 is regular, of index $J=-J^T$8, and all finite eigenvalues lie in $J=-J^T$9 if and only if

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$00

and

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$01

(Chu et al., 2024). Strict passivity is characterized separately by $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$02 and an explicit positivity condition involving $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$03 and $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$04 (Chu et al., 2024).

The 2025 paper on regularization isolates the rank conditions for output-feedback regularization without requiring full asymptotic stabilization, and emphasizes that all constructions proceed through condensed forms computed using orthogonal transformations, so the resulting algorithms are numerically reliable (Chu et al., 2 Sep 2025). The 2025 output-feedback stabilization paper extends the unknown-$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$05 case and states that, under its two rank conditions, any positive definite feedback matrix $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$06 achieves a regular, impulse-free, asymptotically stable, structure-preserving closed loop (Shi et al., 29 Dec 2025). A plausible implication is that the pH structure makes a classically difficult descriptor feedback problem much more tractable than in the unstructured setting; that contrast is stated explicitly in the stabilization papers (Chu et al., 2024, Shi et al., 29 Dec 2025).

5. Passivity, positive realness, controllability, and realizations

The relation among port-Hamiltonian, passive, positive-real, and KYP-based system classes is highly structured but not fully reversible without additional hypotheses. For any regular descriptor system $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$07,

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$08

(Cherifi et al., 2022). Here positive realness means that

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$09

is analytic for $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$10 and satisfies $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$11 there (Chu et al., 2024, Cherifi et al., 2022). The generalized KYP inequality is

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$12

(Cherifi et al., 2022).

The converses require hypotheses. If in the KYP inequality the solution $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$13 is invertible, then one can reconstruct a pH representation explicitly (Cherifi et al., 2022). If $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$14 is observable, then any KYP solution $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$15 is invertible; if $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$16, then KYP implies pH (Cherifi et al., 2022). Passivity implies pH under behavioral observability and $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$17 (Cherifi et al., 2022). Positive realness implies passivity under behavioral controllability (Cherifi et al., 2022).

The converse from positive real to port-Hamiltonian becomes exact for minimal descriptor realizations. If $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$18 is regular, completely controllable, completely observable, and positive real, then it is already a pH descriptor system if and only if

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$19

(Chu et al., 2024). If $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$20 fails to be nonnegative, an equivalent pH realization can still be constructed for descriptor systems by a feedthrough shift: find $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$21 such that

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$22

then define

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$23

The new system $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$24 is port-Hamiltonian and has the same transfer function (Chu et al., 2024). In the standard case $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$25, by contrast, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$26 is invariant under similarity, so such feedthrough shifting is not available (Chu et al., 2024).

Controllability and stabilizability also admit structural dimension criteria in the pH descriptor class. For systems

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$27

with $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$28, five pH controllability concepts are relative generic in $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$29, $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$30, and $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$31 precisely under explicit inequalities in $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$32 (Ilchmann et al., 2023). In particular, freely initializable and impulse-controllable are relative generic iff $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$33; behavioral-controllable iff $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$34; and completely- and strongly-controllable iff $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$35 (Ilchmann et al., 2023). The stabilizability criteria have the same dimension pattern, with a non-generic boundary case at $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$36 (Ilchmann et al., 2023).

These results clarify a frequent source of confusion: passivity, positive realness, and port-Hamiltonian structure are closely related, but they are not identical notions. The descriptor setting makes the differences particularly visible because the feedthrough term, algebraic constraints, and index can obstruct converse implications (Cherifi et al., 2022, Chu et al., 2024).

6. Numerical linear algebra, discretization, model reduction, and extensions

Port-Hamiltonian descriptor systems are unusually well aligned with structure-preserving numerical linear algebra. Condensed forms under orthogonal transformations are a recurring tool for regularity tests, index analysis, stabilization, and realization algorithms (Mehrmann et al., 2022, Chu et al., 2 Sep 2025). In the 2025 regularization paper, the reduction of $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$37 uses simultaneous left/right orthogonal transforms $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$38 so that $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$39 acquire a staircase form with nonsingular diagonal subblocks, followed by further orthogonal transformations that isolate the infinite-eigenvalue part into a $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$40 block and a final real Schur decomposition (Chu et al., 2 Sep 2025). The full feedback computation uses only QR, CS-decomposition, SVD, and real-Schur steps, so the algorithm is numerically reliable (Chu et al., 2 Sep 2025).

The same emphasis on structure appears in eigenvalue computations. For the pencil

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$41

the relation

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$42

implies the spectral symmetry $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$43 (Mehrmann et al., 2020). Structured backward-error analysis shows that if a computed spectrum has the correct symmetry, then there exists a nearby pH descriptor system with exactly that eigenstructure, and the corresponding bounds depend on a conditioning constant related to the stability radius $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$44 (Mehrmann et al., 2020). A large stability radius means that imposing the pH structure causes only small growth in backward error, whereas near-singular or marginally stable systems are more ill-conditioned (Mehrmann et al., 2020).

Time discretization can be carried out in a structure-preserving manner. For nonlinear pHDAEs

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$45

collocation and discrete-gradient methods preserve a discrete power balance (Mehrmann et al., 2019). In particular, Gauss-type collocation yields

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$46

and the midpoint rule reproduces the continuous one-step dissipation law exactly in the quadratic case (Mehrmann et al., 2019). There is also a discrete-time dissipative pH descriptor formulation for completely causal scattering-passive systems,

$E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$47

with discrete-time KYP characterizations and equivalence to scattering passivity and bounded realness under the stated assumptions (Cherifi et al., 2023).

Model reduction has been developed in a structure-preserving way. LQG balanced truncation for pH descriptor systems uses two generalized algebraic Riccati equations, one of them modified by the addition of $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$48, and preserves the pH structure of the reduced model (Breiten et al., 2021). The reduced-order error is bounded through normalized right coprime factorizations: $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$49 and the choice of Hamiltonian can be improved by replacing $E\dot x=(J-R)Qx+Bu,\ y=B^{T}Qx$50 with an extremal solution of a generalized KYP inequality (Breiten et al., 2021). The paper reports that this can significantly improve reduced-order accuracy while retaining passivity and pH form (Breiten et al., 2021).

Applications span electrical circuits, multi-body systems, fluid and gas networks, wave networks, damped beam models, seepage, and nanorods (Beattie et al., 2017, Mehrmann et al., 2022, Bendimerad-Hohl et al., 2024). The 2017 foundational pHDAE paper emphasizes high-index examples such as RLC networks, gas-pipeline wave networks, and constrained multibody systems, and shows how hidden constraints can be exposed while preserving the pH structure (Beattie et al., 2017). The 2022 survey describes the same framework as adequate for simulation, control, and model reduction, and closes with open problems and research topics deserving further attention (Mehrmann et al., 2022).

A plausible synthesis is that pH descriptor systems are not merely a special representation of DAEs, but a structural class in which regularization, stability analysis, feedback design, discretization, and reduction can all be posed in a way that preserves the same energy and passivity identities from modeling through computation (Beattie et al., 2017, Mehrmann et al., 2022, Chu et al., 2 Sep 2025).