Singular Port-Hamiltonian Systems Beyond Passivity

TL;DR

Problem addressed: a class of singular port-Hamiltonian systems with an ellipsoidal singular set $\\mathcal{S}=\\{x:\nabla x^T Q x = 1\\}$ challenges standard passivity-based stability for grid-forming control. Approach: analyze energy shaping with $H(x)=\tfrac12 \\sigma(x)^2$, where $\\sigma(x)=\tfrac12(x^T Q x-1)$, show dissipativity outside $\\mathcal{S}$, establish forward-invariance of $\\mathcal{S}$ and interconnection-induced convergence to $\\mathcal{S}$, and propose two implementable approximations (finite jump and linear boundary layer) that preserve key dissipativity properties. Key contributions: (i) outside $\\mathcal{S}$ the system is passive with storage $H$, (ii) trajectories converge to $\\mathcal{S}$ under interconnection with passive loads, (iii) two practical approximations retain dissipativity (one passive outside $\\mathcal{S}$, one cyclo-dissipative) and mitigate singularity, and (iv) numerical experiments illustrate finite-time convergence to a non-equilibrium steady state on $\\mathcal{S}$. Significance: provides rigorous stability guidance and practical control design for inverter-dominated grids where energy pumping occurs across a singular set, connecting passivity concepts to non-passive, energy-injecting behavior.

Abstract

In this paper, we study a class of port-Hamiltonian systems whose vector fields exhibit singularities. A representative example of this class has recently been employed in the power electronics literature to implement a grid-forming controller. We show that, under certain conditions, these port-Hamiltonian systems, when interconnected with passive systems, converge to a prescribed non-equilibrium steady state. At first glance, the apparently passive nature of the port-Hamiltonian system seems incompatible with the active power injection required to sustain this non-equilibrium condition. However, we demonstrate that the discontinuity inherent in the vector field provides the additional energy needed to maintain this operating point, indicating that the system is not globally passive. Moreover, when the discontinuity is replaced by a continuous approximation, the resulting system becomes cyclo-dissipative while still capable of supplying the required power.

Figures (4)

• Figure 1: The sets $\mathcal{S}$ (black) and $\mathcal{M}$ (orange), for $Q=I$, $M=3$, and $n=2$.
• Figure 2: The phase portraits from Example \ref{['ex:static_passive']}. The state converges quickly to $\mathcal{S}$ for different initial states, under two different interconnections $K$.
• Figure 3: State trajectories $x_1(t)$ from Example \ref{['ex:static_passive']}. The state converges quickly to the desired frequency $\omega_0=2\pi$ rad/s of amplitude $1$, under two different interconnections $K$.
• Figure 4: The phase portraits from Example \ref{['ex:dynamic_passive']}. The state converges to the desired point on $\mathcal{S}$ for different dissipations, but leaves the vicinity of $\mathcal{S}$ briefly.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Proposition 1
• proof
• Theorem 1
• Theorem 2
• Proposition 2
• Remark 3
• Proposition 3
• Example 1: Frequeny Tracking with Static Passive Load