The difference between port-Hamiltonian, passive and positive real descriptor systems

The paper proves that a descriptor system Σ is port-Hamiltonian if and only if there exists Q with E^H Q = Q^H E ≥ 0 solving the generalized KYP inequality and satisfying ker Q ⊆ ker A ∩ ker C (Proposition 3.3), showing the forward KYP factorization K(Q) = 2·diag(Q^H, I)·W·diag(Q, I) for pH systems (equations (11)–(12)) and then deriving a pH representation from a KYP solution by constructing a dissipative Θ (and splitting Θ = Γ − W with Γ = −Γ^H, W ≥ 0) . The model’s solution establishes the same equivalence but uses a lifting/factorization lemma K(Q) = 2·T^H W T with T = diag(Q, I) (under ker T ⊆ ker K) to obtain W ≥ 0 directly, then builds P,S,G and constructs J = −J^H with JQ = A + RQ. Both proofs are correct; they differ mainly in how the reverse direction factors the KYP inequality.