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Multiple Control Functionals for Interconnected Time-Delay Systems

Zhuo-Rui Pan, Wei Ren, Xi-Ming Sun

TL;DR

This work addresses safe stabilization of interconnected time-delay systems by introducing multiple control Lyapunov and barrier functionals that interact through a distributed small-gain structure. It develops two complementary controller designs: an optimization-based (implicit) method that solves distributed QPs to enforce safety while following a stabilizing nominal law, and a sliding-mode (explicit) method that constructs a manifold from the functionals for direct controller implementation. The approach extends barrier-function concepts to time-delay interconnections and demonstrates its effectiveness on a four-robot reach-avoid task with delays. The results offer scalable, distributed strategies for guaranteeing stability and safety in cyber-physical systems with communication delays and inter-subsystem couplings.

Abstract

Safety is essential for autonomous systems, in particular for interconnected systems in which the interactions among subsystems are involved. Motivated by the recent interest in cyber-physical and interconnected autonomous systems, we address the safe stabilization problem of interconnected systems with time delays. We propose multiple control Lyapunov and barrier functionals for the stabilization and safety control problems, respectively. In order to investigate the safe stabilization control problem, the proposed multiple control functionals are combined together via two methods: the optimization-based method and the sliding mode based method. The resulting controllers can be of either explicit or implicit forms, both of which ensure the safe stabilization objective of the whole system. The derived results are illustrated via a reach-avoid problem of multi-robot systems.

Multiple Control Functionals for Interconnected Time-Delay Systems

TL;DR

This work addresses safe stabilization of interconnected time-delay systems by introducing multiple control Lyapunov and barrier functionals that interact through a distributed small-gain structure. It develops two complementary controller designs: an optimization-based (implicit) method that solves distributed QPs to enforce safety while following a stabilizing nominal law, and a sliding-mode (explicit) method that constructs a manifold from the functionals for direct controller implementation. The approach extends barrier-function concepts to time-delay interconnections and demonstrates its effectiveness on a four-robot reach-avoid task with delays. The results offer scalable, distributed strategies for guaranteeing stability and safety in cyber-physical systems with communication delays and inter-subsystem couplings.

Abstract

Safety is essential for autonomous systems, in particular for interconnected systems in which the interactions among subsystems are involved. Motivated by the recent interest in cyber-physical and interconnected autonomous systems, we address the safe stabilization problem of interconnected systems with time delays. We propose multiple control Lyapunov and barrier functionals for the stabilization and safety control problems, respectively. In order to investigate the safe stabilization control problem, the proposed multiple control functionals are combined together via two methods: the optimization-based method and the sliding mode based method. The resulting controllers can be of either explicit or implicit forms, both of which ensure the safe stabilization objective of the whole system. The derived results are illustrated via a reach-avoid problem of multi-robot systems.
Paper Structure (13 sections, 3 theorems, 23 equations, 2 figures)

This paper contains 13 sections, 3 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Interconnected Time-Delay Systems
  4. Stabilization Control
  5. Safety Control
  6. Multiple Control Functionals
  7. Multiple Control Lyapunov Functionals
  8. Multiple Control Barrier Functionals
  9. Safe Stabilization Controller Design
  10. Optimization-based Design
  11. Sliding Mode based Design
  12. Numerical Results
  13. Conclusion

Key Result

Theorem 1

If the system $\mathcal{S}$ admits MCLFs and satisfies the DSCP, then the closed-loop system is GAS under the continuous controller designed below: where $i\in\mathcal{N}$, $\mathfrak{a}_{i}(\phi):=L_{f_{i}}V_{i1}(\phi)+D^{+}V_{i2}(\phi_{i})+\rho_{i}(V_{i}(\phi_{i}))-\sum_{j\in\mathcal{N}}\gamma_{ij}(V_{j}(\phi_{j}))$, $\mathfrak{b}_{i}(\phi):=L_{g_{i}}V_{i1}(\phi)$, and

Figures (2)

  • Figure 1: Illustration of the position trajectories of all robots via the optimization based control design. The dark grey regions are the obstacles, the initial positions are the dots, and the terminal positions are the crosses.
  • Figure 2: Illustration of the position trajectories of all robots via the sliding mode based control design. The dark grey regions are the obstacles, the initial positions are the dots, and the terminal positions are the crosses.

Theorems & Definitions (13)

  • Remark 1
  • Definition 1
  • Definition 2: Di2017robustification
  • Definition 3: Di2017robustification
  • Definition 4
  • Remark 2
  • Definition 5
  • Theorem 1
  • Definition 6
  • Theorem 2
  • ...and 3 more