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Port-Transversal Barriers: Graph-Theoretic Safety for Port-Hamiltonian Systems

Chi Ho Leung, Philip E. Paré

Abstract

We study port-Hamiltonian systems with energy functions that split into local storage terms. From the interconnection and dissipation structure, we construct a graph on the energy compartments. From this graph, we show that the shortest-path distance from a constrained compartment to the nearest actuated one gives a lower bound on the relative degree of the corresponding safety constraint. We also show that no smooth static feedback can reduce it when no path exists. When the relative degree exceeds one and the immediate graph neighbors of the constrained compartment is connected to at least one input port, we reshape the constraint by subtracting their shifted local storages, producing a candidate barrier function of relative degree one. We then identify sufficient regularity conditions that recover CBF feasibility under bounded inputs. We validate the framework on an LC ladder network, where the enforceability of a capacitor charge constraint depends only on the input topology.

Port-Transversal Barriers: Graph-Theoretic Safety for Port-Hamiltonian Systems

Abstract

We study port-Hamiltonian systems with energy functions that split into local storage terms. From the interconnection and dissipation structure, we construct a graph on the energy compartments. From this graph, we show that the shortest-path distance from a constrained compartment to the nearest actuated one gives a lower bound on the relative degree of the corresponding safety constraint. We also show that no smooth static feedback can reduce it when no path exists. When the relative degree exceeds one and the immediate graph neighbors of the constrained compartment is connected to at least one input port, we reshape the constraint by subtracting their shifted local storages, producing a candidate barrier function of relative degree one. We then identify sufficient regularity conditions that recover CBF feasibility under bounded inputs. We validate the framework on an LC ladder network, where the enforceability of a capacitor charge constraint depends only on the input topology.

Paper Structure

This paper contains 22 sections, 7 theorems, 45 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Results
  4. Port Transversality
  5. Compartmental Structure and Graphs
  6. Port Nodes, Barrier Nodes, and Transversality
  7. Relative Degree from Graph Distance
  8. Port-Transversal Barrier Synthesis
  9. Port-Energy Shaping of the Barrier
  10. Port-Transversal Barrier Feasibility
  11. Strong Structural Condition
  12. Numerical Simulations
  13. Constraint Enforcement
  14. Safe Set Geometry
  15. Conclusion
  16. ...and 7 more sections

Key Result

Lemma 3.1.1

Under Assumption ass:structural, the summands of $L_f^k\varphi$ can be grouped into partial sums indexed by walks of length at most $k$ from $\mathcal{B}$ in $\mathcal{G}(A)$, where each partial sum depends only on compartments visited by its walk. In particular, $\operatorname{csupp}(L_f^k \varphi)

Figures (7)

  • Figure 1: Influence graphs $\mathcal{G}(A)$ of three port-Hamiltonian systems. (Left) Series RLC: nodes $C$ and $L$ represent the capacitor and inductor energy compartments, and the edge $\{C, L\}$ encodes the power exchange between the capacitor and inductor. (Center) LC Ladder with two voltage sources. (Right) Serial Mass-Spring: the port enters only at mass $m_2$ with potential and kinetic storage on springs $s_1, s_2$ and masses $m_1, m_2$ with input applying on the momentum of mass $m_2$.
  • Figure 2: Dual-input LC ladder network that induces the influence graph in Fig. \ref{['fig:influence_graphs']} (Center).
  • Figure 3: Port-transversal barrier for a charge constraint $q_{\max} = 1.5$ on the series RLC circuit. The safety specification $\varphi(x) = q_{\max} - q$ (dashed red) defines the allowable set $\mathcal{A} = \{q \leq q_{\max}\}$. The port-transversal barrier $h_\gamma(x) = \varphi(x) - \frac{1}{\gamma}\,\bar{H}_L(\phi)$ reshapes the safety specification by subtracting the shifted inductor storage $\bar{H}_L = \phi^2/(2L)$ from the insulating blanket $\partial\mathcal{B} = \{L\} = \mathcal{P}$ (blue contour). As $\gamma$ increases, the energy penalty shrinks and the safe set $\mathcal{S}_\gamma = \{h_\gamma \geq 0\}$ expands toward $\mathcal{A}$, consistent with the monotonicity $\mathcal{S}_{\gamma_1} \subseteq \mathcal{S}_{\gamma_2}$ for $\gamma_1 \leq \gamma_2$ (Lemma \ref{['lem:pt-synthesis']} (\ref{['item:safe-set']})). The black dashed line is the degeneracy set $\mathcal{Z}_{\partial \mathcal{B}}$, and the black trajectory shows the driven system under nominal input with the black dot the final state.
  • Figure 4: Charge constraint enforcement on the LC ladder. (Left) Capacitor charge $q_1(t)$: the dual-input (port-insulated) filter respects the bound $q_{\max} = 0.8$ (dashed red); the single-input filter violates it repeatedly. (Right) Barrier value $h_\gamma(t)$: the dual-input barrier stays nonnegative; the single-input barrier drops below zero (red shading), indicating infeasibility of the CBF condition.
  • Figure 5: Control authority and phase portrait. (Left) The control authority $\|L_G h_\gamma\|$: the dual-input system never loses input sensitivity, while the single-input authority drops to zero (red dots mark infeasible steps). (Right) Phase portrait $(q_1,\phi_1)$ at the $\phi_2 = 0$ slice: the dual-input trajectory (blue) stays within the safe set; the single-input trajectory (orange) crosses the constraint boundary.
  • ...and 2 more figures

Theorems & Definitions (29)

  • Definition 1: Power flow graph
  • Definition 2: Influence graph
  • Example 1: Dual-inputs LC ladder network
  • Definition 3: Compartmental support
  • Definition 4: $k$-hop neighborhood
  • Definition 5: Relative degree
  • Definition 6: Port transversality
  • Lemma 3.1.1: Support propagation
  • proof
  • Theorem 1: Relative degree from graph distance
  • ...and 19 more