Tube-Based Safety for Anticipative Tracking in Multi-Agent Systems

Abstract: A tube-based safety framework is presented for robust anticipative tracking in nonlinear Brunovsky multi-agent systems subject to bounded disturbances. The architecture establishes robust safety certificates for a feedforward-augmented ancillary control policy. By rendering the state-deviation dynamics independent of the agents' internal nonlinearities, the formulation strictly circumvents the restrictive Lipschitz-bound feasibility conditions otherwise required for robust stabilization. Consequently, this structure admits an explicit, closed-form robust positively invariant (RPI) tube radius that systematically attenuates the exponential control barrier function (eCBF) tightening margins, thereby mitigating constraint conservatism while preserving formal forward invariance. Within the distributed model predictive control (MPC) layer, mapping the local tube radii through the communication graph yields a closed-form global formation error bound formulated via the minimum singular value of the augmented Laplacian. Robust inter-agent safety is enforced with minimal communication overhead, requiring only a single scalar broadcast per neighbor at initialization. Numerical simulations confirm the framework's efficacy in safely navigating heterogeneous formations through cluttered environments.

• The paper introduces a feedforward compensation method that decouples follower nonlinearities from the leader’s dynamics, eliminating conservative Lipschitz constraints.
• It derives an explicit, closed-form tube radius to ensure non-conservative safety margins in distributed MPC under bounded disturbances.
• The work establishes a network-indexed formation tracking error bound that guarantees robust performance and minimal communication overhead in multi-agent systems.

Tube-Based Safety for Anticipative Tracking in Multi-Agent Systems: Technical Summary and Research Context

Problem Setting and Limitations of Prior Methods

This work addresses robust safety in nonlinear leader-follower multi-agent systems (MAS) with high-order Brunovsky-type agent dynamics, operating under bounded disturbances, hard safety (collision/obstacle) constraints, and distributed information architectures. In standard distributed MPC (DMPC) architectures, safety enforcement under bounded disturbances is typically handled either via conservative constraint tightening (e.g., tube-based RMPC [Mayne, 2005]) or control barrier functions (CBFs), but architectural limitations emerge for high-relative-degree nonlinearities and in fully distributed multi-agent scenarios. Previous robust tube-based MPC/DMPC approaches for MASs require treating all unknown nonlinearities as lumped disturbances, which implicitly restricts robust stabilization feasibility to systems with small Lipschitz constants—even when the true agent dynamics are exactly known or partially cancelable by model-based feedforward [RIVERSO20142179, koulong2025wc].

Technical Contributions

1. Deviation Dynamics Decoupling via Feedforward Compensation

The framework exploits explicit knowledge of both leader and follower dynamics to design a feedforward-augmented ancillary control policy. By introducing a control law that compensates the follower's inherent nonlinearities with a feedforward of the leader’s nonlinear dynamics, the deviation dynamics for each follower become independent of the follower’s own unknown nonlinearity. The only remaining disturbances are the mismatch between actual and nominal leader trajectories (propagated through the communication graph) and bounded external perturbations. This structural decoupling eliminates the restrictive Lipschitz-dependent feasibility condition present in standard robust invariant set (RPI) tube constructions, which otherwise would be necessary for ensuring non-conservative robustification.

2. Explicit Closed-Form Tube Radius and Safety Margin Tightening

The core technical result is an explicit, closed-form expression for the RPI tube radius of each follower's deviation dynamics. Compared to prior general-purpose methods, which embed the nonlinear term's Lipschitz constant in the denominator (making the tube arbitrarily large for highly nonlinear systems), the feedforward compensation removes the dependence on the nonlinear follower dynamics. The resulting tube radius for each follower is given by

$$r_i \geq \frac{2 \lambda_{\max}(P_i)^{3/2} \|G\|\,\bar{w}^i_{\mathrm{eff}}} {\lambda_{\min}(Q_i)}$$

where $\bar{w}^i_{\mathrm{eff}}$ explicitly captures only residual disturbances. Absence of the Lipschitz constant in this bound is a crucial structural innovation. The tube-based tightening is then directly applied to the exponential CBF constraints required for both inter-agent collision and obstacle avoidance, guaranteeing that constraint satisfaction of the nominal MPC plan implies safety of the true disturbed system trajectory.

3. Network-Indexed Global Formation Error Bound

By leveraging the communication graph structure, the paper rigorously derives a closed-form upper bound on the collective formation tracking error in terms of the minimum singular value of the formation’s augmented graph Laplacian (including leader-to-follower and inter-follower links). This result formalizes how individual per-agent tube radii and the network topology together bound the worst-case deviation from the desired formation manifold, offering system-level performance guarantees not present in previous methods.

4. Communication Architecture

The method maintains minimal communication overhead: at initialization, each agent broadcasts its computed tube radius (a scalar) to its neighbors, which suffices for the local computation of the necessary safety tightening terms. At runtime, the only per-step information exchange is the one-step-lagged neighbor control plan, consistent with distributed non-iterative DMPC schemes [DunbarMurray2006, Shorinwa2024].

Numerical Results

In the presented numerical experiments, the proposed tube-based anticipative tracking framework demonstrates significantly smaller steady-state formation errors and minimum feasible clearances when compared with general-purpose robust eCBF-tightened DMPC [koulong2025wc]. The reduction in conservatism is quantitatively attributed to the structural removal of the follower’s nonlinearities from the RPI tube construction, allowing the DMPC to focus on the formation objective rather than expending control authority to hedge against artificially inflated disturbance estimates. Empirical safety is maintained throughout, but with noticeably less aggressive (intrusive) safety margins.

Theoretical and Practical Implications

Theoretical:

• Feasibility Region Expansion: By removing the Lipschitz constant from the robust stabilization constraint, the method notably expands the class of nonlinear MAS for which formal robustification is tractable, including systems with severe nonlinearities and/or minimal model uncertainty.
• Explicit Network Performance Bound: The formation tracking bound elucidates the interplay between local safety margins and the global communication topology, analytically connecting invariant set geometry with the network spectral gap.

Practical:

• Reduced Controller Authoritative Conflict: The controller can prioritize the formation objective instead of allocating authority for overly conservative constraint tightening.
• Minimal Communication Complexity: Only scalar tube radii and neighbor control plan predictions are exchanged, ensuring scalability.

Relation to Existing Work

This architecture systematically tightens and improves upon existing robust DMPC and CBF-MPC approaches [Mayne2005RMPC, XuAmes2015RobustCBF, Jankovic2018RobustCBF, Zeng2021MPCDiscreteCBF, Liu2023IterativeMPCDHOCBF, Dai2017DMPCFormationSurveyLike, koklong2025wc], overcoming key computational and feasibility limitations in existing distributed safety-critical MAS controllers.

Future Research Directions

The paper identifies several research frontiers:

• Switching Topologies: Extensions to autonomous or event-triggered switching communication graphs, with robustification under dynamic reconfiguration and dwell-time constraints [LI2020].
• Communication Delays: Integration of delay-compensating observers or predictive protocols in the context of distributed robust safety [ZHANG2024, Xu2023].
• Adaptive/Online Tube Tightening: Adaptive or data-driven tube size reduction could further mitigate conservatism at the expense of communication frequency increase.
• Physical Validation: Practical deployment on hardware multi-robot platforms to demonstrate effectiveness under real disturbances, sensor noise, and network uncertainties.

Conclusion

The framework establishes a tube-based safety architecture for anticipative leader-follower tracking in distributed nonlinear MAS. By leveraging explicit model compensation and network-theoretic analysis, the method yields non-conservative, closed-form safety bounds and system-level performance guarantees. The approach is theoretically rigorous, practically implementable, and generalizable to high-order MAS with heterogeneous nonlinearities, providing a solid foundation for scalable, robust, and safe distributed control under uncertainty.

---

References

• Mayne, D. Q. et al., "Robust model predictive control of constrained linear systems with bounded disturbances," Automatica, 2005.
• Koulong, A., Pakniyat, A., "Robust Multi-Agent Safety via Tube-Based Tightened Exponential Barrier Functions," (Koulong et al., 26 Oct 2025), 2025.
• Dunbar, W. B. and Murray, R. M., "Distributed receding horizon control for multi-vehicle formation stabilization," Automatica, 2006.
• Xu, X., Ames, A. D., "Robustness of Control Barrier Functions for Safety Critical Control," IFAC-PapersOnLine, 2015.
• Jankovic, M., "Robust control barrier functions for constrained stabilization of nonlinear systems," Automatica, 2018.
• RIVERSO, S. et al., "Plug-and-play model predictive control based on robust control invariant sets," Automatica, 2014.
• Shorinwa, O., Schwager, M., "Distributed Model Predictive Control via Separable Optimization in Multiagent Networks," IEEE TAC, 2024.
• Zeng, J. et al., "Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function," ACC, 2021.
• Dai, L. et al., "Distributed MPC for formation of multi-agent systems with collision avoidance and obstacle avoidance," J. Franklin Institute, 2017.
• LI, K. et al., "Distributed model predictive control of multi-vehicle systems with switching communication topologies," Transp. Res. Part C, 2020.