Source-linked AI summary
Networked Admissibility-Preserving Control for Directed Safe Coordination
Abhinav Sinha, Lohitvel Gopikannan, Shashi Ranjan Kumar
TL;DR
Safety-critical coordination must preserve a moving output corridor and asymmetric physical-input bounds while actuator realization evolves dynamically over a directed network. The paper combines APIR with a logarithmic barrier coordinate to obtain an exact cascade, and proves complete safe operation with bounded commands and exponential consensus on compatible compact initial sets.
Problem
Safe networked coordination must account for physical-input realization transients while preserving moving output corridors and heterogeneous asymmetric actuator bounds on rooted directed graphs.
Method
The paper combines APIR with a logarithmic barrier coordinate to realize directed consensus through an exact network–APIR cascade.
Results
For every compatible compact initial set, the closed loop has a unique complete solution, preserves corridor and actuator invariance with uniform margins, keeps commands bounded, and achieves exponential consensus.
Takeaways & Limitations
The framework characterizes how APIR transients affect directed agreement while providing directional certificates for asymmetric actuator limits and safe collective motion under partial pinning.
Abstract
from arXiv · showhide
This paper addresses safety-critical coordination for scalar agents whose distributed commands are implemented through constrained physical-input dynamics. Agents communicate over a fixed weighted digraph with a directed spanning tree, while their outputs must remain inside a common moving safety corridor and their realized inputs must satisfy heterogeneous asymmetric bounds. We propose a networked Admissibility-Preserving Control (APC) architecture in which an Admissibility-Preserving Input Realization (APIR) governs physical inputs and a logarithmic barrier coordinate represents the safety corridor. The synthesis yields an exact cascade in which exponentially decaying realization errors drive nonsymmetric consensus dynamics. For every compatible compact initial set, the closed-loop system admits a unique complete solution, renders the moving corridor and actuator intervals forward invariant with uniform margins, keeps commands bounded, and achieves exponential consensus. We derive direction-specific sufficient conditions under which positive and negative control demands remain within their corresponding actuator limits. The analysis yields a closed-form barrier-coordinate limit determined by the left Perron vector and initial APIR mismatch. Under strong connectivity and the stated gain and compatibility conditions, partial pinning propagates a constant barrier reference from a nonempty informed subset and assigns the induced safety corridor trajectory. A non-weight-balanced example illustrates the directional certificate and predicted collective motion.
I. INTRODUCTION
The paper targets safe leaderless coordination over rooted, generally unbalanced digraphs when physical inputs are constrained dynamic states with heterogeneous asymmetric bounds. It proposes a directed networked APC architecture that preserves the moving output corridor and actuator intervals during realization transients while characterizing their effect on agreement.
- Motivation: Physical-input realization transients can consume output margin and shift the directed network’s eventual agreement value before realization errors decay.This motivates treating actuator realization as part of the closed-loop dynamics rather than only constraining an algebraic command.
- Motivation: Existing networked safety designs do not simultaneously preserve a common moving output corridor and heterogeneous asymmetric actuator intervals for leaderless consensus over rooted, generally unbalanced digraphs.The gap concerns both directed-network imbalance and physical actuator realization as a dynamic state.
- Contribution: The proposed APC architecture combines APIR, a logarithmic barrier coordinate, and an exact cascade in which exponentially decaying realization errors drive rooted-digraph consensus dynamics.The construction supplies direction-specific compatibility conditions and a transient correction to the directed agreement value.
- Contribution: Unlike funnel formulations, the compatibility certificate enforces designer-specified asymmetric actuator intervals without widening the prescribed moving output corridor.The architecture extends networked APC from undirected graphs with global reference access to leaderless rooted digraphs with directional certification.
- Contribution: Under strong connectivity, partial pinning assigns a safe collective trajectory from a nonempty informed subset.The supplied passage states that the extension propagates a constant barrier reference and assigns the induced safety corridor trajectory.
II. PROBLEM FORMULATION
The formulation models scalar agents on a fixed weighted digraph with constrained physical inputs and outputs inside a common moving corridor. A logarithmic barrier coordinate and APIR-based dynamics convert the safety problem into directed consensus while seeking completeness, invariance, bounded commands, and agreement on compatible compact initial sets.
- Graph model: The fixed weighted digraph is assumed to contain a directed spanning tree, yielding a topology-dependent left Perron vector and collective-mode projection.The projection P extracts the collective mode, while Π extracts disagreement.
- Agent and actuator model: Each agent retains its physical input as a constrained dynamic state and generates a dedicated command that drives the APIR.The prescribed actuator set is heterogeneous and asymmetric, with lower and upper bounds straddling zero.
- APIR realization: APIR dynamics point inward at both actuator boundaries, generating a compact invariant interior subset under bounded commands.The later closed-loop analysis supplies the required command bound and direction-dependent invariant actuator subset.
- Barrier-coordinate transformation: The logarithmic barrier coordinate maps the moving corridor to an unconstrained coordinate and produces dynamics with input gain h(t)/(qℓ,i qᵤ,i) plus corridor-motion drift.Its inverse map is logistic in the barrier coordinate, and the coordinate diverges at the corridor boundaries.
- Problem statement: The problem seeks distributed commands that keep every output in the common corridor and every physical input in its asymmetric actuator interval while achieving consensus.The formulation also requires complete solutions for all initial conditions in the chosen compact set and bounded commands.
- Compatibility and scope: Finite actuator authority makes the certificate regional: only compact admissible state–actuator subsets whose graph-induced motion and corridor kinematics fit available authority are certifiable.APC compatibility verifies that APIR can realize those motions within the available bounds.
III. MAIN RESULTS
The APC synthesis creates an exact network–APIR cascade whose compatibility certificate preserves moving-corridor and actuator safety while guaranteeing complete, bounded, exponentially convergent coordination. The analysis also characterizes the directed-network agreement shift and extends the result to partial pinning.
- Exact network–APIR cascade: The exact cascade is ż = −kLz + e, ė = −Ce, separating directed consensus dynamics from exponentially decaying APIR realization errors.This cascade is the central APC synthesis mechanism and preserves the rooted-digraph geometry.
- Safety and convergence: Under the one-sided APC compatibility conditions, every initial condition in K yields a unique complete solution with invariant output and actuator constraints, bounded commands, and exponential consensus.Strict output and actuator margins prevent boundary contact and allow continuation for all forward time.
- Safety and convergence: The realization error preserves its initial sign and decays as ∥e(t)∥ ≤ E_K e^(−c_min t), while transformed-state bounds certify a uniform interior output margin m_K.These estimates support both barrier-map conditioning and global safety certification.
- Directional actuator certification: Direction-specific compatibility inequalities match positive and negative control demands to their corresponding asymmetric actuator limits, reducing conservatism relative to a symmetric inner-bound certificate.The certificate treats each actuator direction separately rather than imposing a common symmetric bound.
- Agreement value: The directed-network agreement value equals the standard left-Perron projection when e_0 = 0, but a nonzero APIR mismatch adds a signed asymptotic shift.For a rooted digraph, the root strongly connected component and its APIR transients determine the agreement value.
- Partial pinning: With strong connectivity, partial pinning assigns a constant barrier reference from a nonempty informed subset and induces a safe collective corridor trajectory without requiring π or graph eigenvalues online.The pinned result also provides exponential convergence under the stated gain and compatibility conditions.
IV. SIMULATIONS
Across strongly connected, rooted spanning-tree, and partially pinned scenarios, the proposed network APIR keeps agents within the moving safety corridor and realized inputs within certified bounds while achieving consensus.
- Simulation setup: The simulations use heterogeneous asymmetric actuator intervals and time-varying corridor bounds across five single-integrator agents.The strongly connected case specifies distinct actuator limits and a moving corridor defined by time-varying center and half-widths.
- Strongly connected digraph: All five agents remain within the moving safety corridor and converge to the common agreement trajectory in the strongly connected scenario.The commanded signals remain uniformly bounded, and all actuator compatibility margins are strictly positive.
- Strongly connected digraph: The realized physical inputs stay within certified bounds even when commanded signals exceed the maximum possible actuator limits.This numerically demonstrates APIR efficacy and validates the directional compatibility certificate.
- Directed spanning tree: The spanning-tree scenario also achieves safe consensus inside the moving corridor, with bounded commands and realized inputs remaining within certified bounds.All five actuator compatibility margins are strictly positive, and the results confirm that strong connectivity and weight balance are unnecessary.
- Partial pinning: Under partial pinning, all five agents converge to the physical trajectory generated by the pinned barrier reference, including uninformed agents reached through directed exchange.The realized actuator inputs remain within the analytically certified bounds.
V. CONCLUSIONS
The paper formulates directed safe consensus as a networked APC problem combining constrained physical-input dynamics with barrier-coordinate consensus. Its guarantees include safety margins, bounded commands, complete solutions, directed-network agreement characterization, and partial-pinning trajectory assignment within a regional compatibility certificate.
- Conclusion: APIR retains physical input as a constrained dynamic state while the distributed synthesis realizes barrier-coordinate consensus through admissible input dynamics.The exact cascade preserves rooted-digraph geometry and quantifies APIR-transient corrections to the agreement value.
- Conclusion: The compatibility certificate establishes uniform output margins, direction-specific input bounds, a positive APIR gain margin, bounded commands, and complete solutions.Its regional character reflects finite actuator authority relative to initial transformed dispersion and corridor motion.
- Conclusion: Partial pinning assigns the safe collective trajectory without broadcasting a global reference.The conclusion places this result within the directed safe-consensus formulation.
- Conclusion: Extensions to switching rooted digraphs, vector outputs, uncertain APIR parameters, and sampled neighbor exchange require additional robustness and hybrid arguments.These extensions are identified as future work.