Early-Terminable Energy-Safe Iterative Coupling for Parallel Simulation of Port-Hamiltonian Systems

Abstract

Parallel simulation and control of large-scale robotic systems often rely on partitioned time stepping, yet finite-iteration coupling can inject spurious energy by violating power consistency--even when each subsystem is passive. This letter proposes a novel energy-safe, early-terminable iterative coupling for port-Hamiltonian subsystems by embedding a Douglas--Rachford (DR) splitting scheme in scattering (wave) coordinates. The lossless interconnection is enforced as an orthogonal constraint in the wave domain, while each subsystem contributes a discrete-time scattering port map induced by its one-step integrator. Under a discrete passivity condition on the subsystem time steps and a mild impedance-tuning condition, we prove an augmented-storage inequality certifying discrete passivity of the coupled macro-step for any finite inner-iteration budget, with the remaining mismatch captured by an explicit residual. As the inner budget increases, the partitioned update converges to the monolithic discrete-time update induced by the same integrators, yielding a principled, adaptive accuracy--compute trade-off, supporting energy-consistent real-time parallel simulation under varying computational budgets. Experiments on a coupled-oscillator benchmark validate the passivity certificates at numerical roundoff (on the order of 10e-14 in double precision) and show that the reported RMS state error decays monotonically with increasing inner-iteration budgets, consistent with the hard-coupling limit.

Figures (6)

• Figure 1: Port-based view of a pH interconnection. Storage $\mathcal{S}$, dissipation $\mathcal{R}$, control $\mathcal{C}$, and interaction $\mathcal{J}$ exchange power through port variables $(e,f)$ and are constrained by a Dirac structure $\mathcal{D}$, representing an ideal lossless interconnection ($e^\top f=0$).
• Figure 2: Parallel iterative coupling of port-Hamiltonian subsystems. A macro-step $n$ performs parallel subsystem integration; an inner interface iteration enforces an ideal lossless interconnection $\mathcal{D}$. The update is energy-safe for any finite number of inner iterations and converges to the monolithic update as the inner iterations increase.
• Figure 3: Schematic of the two-oscillator benchmark.
• Figure 4: Trajectories $(q_1,q_2)$ for different inner-iteration budgets $K_n$ and the monolithic reference.
• Figure 5: Two-oscillator certificates versus inner-iteration budget $K_n$. (a) FNE certificate: worst-case numerically evaluated margins for Condition \ref{['ass:fne_portmap']}. (b) Discrete-passivity and augmented-storage diagnostics: positive-part summaries $(\cdot)_+$ of the discrete-passivity residual \ref{['eq:discrete_passivity_scattering']} and the augmented-storage residual associated with \ref{['eq:augmented_storage_inequality']}.

Theorems & Definitions (14)

• Remark 1: Closed-form coupling projection
• Remark 2: Connection to explicit interfaces
• Remark 3: Passivity-preserving integrators satisfy Condition
• Proposition 1: A sufficient $\gamma$-rule for FNE (impedance bound)
• proof
• Remark 4: $\gamma$ as a tuning parameter
• Remark 5: Monotonicity of the linear coupling operator
• Theorem 1: Energy safety under finite inner iterations
• proof
• Lemma 1: Inner-loop convergence to the monolithic interface