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From Data to Global Asymptotic Stability of Unknown Large-Scale Networks with Provable Guarantees

Mahdieh Zaker, Amy Nejati, Abolfazl Lavaei

TL;DR

This work addresses stabilizing large-scale networks with unknown dynamics by proposing a compositional data-driven framework that yields GAS guarantees. Each unknown subsystem is equipped with an ISS Lyapunov function and local ISS controller derived from a single input–state trajectory, using a data-based representation to bypass explicit model identification. A small-gain–based compositional theorem then constructs a network-level CLF, ensuring GAS with a distributed controller, and SOS-based synthesis scales linearly with the number of subsystems. The method is validated on diverse topologies (fully connected, ring, binary-tree, star, and line Lu networks), showing scalability up to several thousand states. This approach offers a practical, scalable alternative to monolithic model-based stability analysis for unknown, high-dimensional networks.

Abstract

We offer a compositional data-driven scheme for synthesizing controllers that ensure global asymptotic stability (GAS) across large-scale interconnected networks, characterized by unknown mathematical models. In light of each network's configuration composed of numerous subsystems with smaller dimensions, our proposed framework gathers data from each subsystem's trajectory, enabling the design of local controllers that ensure input-to-state stability (ISS) properties over subsystems, signified by ISS Lyapunov functions. To accomplish this, we require only a single input-state trajectory from each unknown subsystem up to a specified time horizon, fulfilling certain rank conditions. Subsequently, under small-gain compositional reasoning, we leverage ISS Lyapunov functions derived from data to offer a control Lyapunov function (CLF) for the interconnected network, ensuring GAS certificate over the network. We demonstrate that while the computational complexity for designing a CLF increases polynomially with the network dimension using sum-of-squares (SOS) optimization, our compositional data-driven approach significantly mitigates it to \emph{linear} with respect to the number of subsystems. We showcase the efficacy of our data-driven approach over a set of benchmarks, involving physical networks with diverse interconnection topologies.

From Data to Global Asymptotic Stability of Unknown Large-Scale Networks with Provable Guarantees

TL;DR

This work addresses stabilizing large-scale networks with unknown dynamics by proposing a compositional data-driven framework that yields GAS guarantees. Each unknown subsystem is equipped with an ISS Lyapunov function and local ISS controller derived from a single input–state trajectory, using a data-based representation to bypass explicit model identification. A small-gain–based compositional theorem then constructs a network-level CLF, ensuring GAS with a distributed controller, and SOS-based synthesis scales linearly with the number of subsystems. The method is validated on diverse topologies (fully connected, ring, binary-tree, star, and line Lu networks), showing scalability up to several thousand states. This approach offers a practical, scalable alternative to monolithic model-based stability analysis for unknown, high-dimensional networks.

Abstract

We offer a compositional data-driven scheme for synthesizing controllers that ensure global asymptotic stability (GAS) across large-scale interconnected networks, characterized by unknown mathematical models. In light of each network's configuration composed of numerous subsystems with smaller dimensions, our proposed framework gathers data from each subsystem's trajectory, enabling the design of local controllers that ensure input-to-state stability (ISS) properties over subsystems, signified by ISS Lyapunov functions. To accomplish this, we require only a single input-state trajectory from each unknown subsystem up to a specified time horizon, fulfilling certain rank conditions. Subsequently, under small-gain compositional reasoning, we leverage ISS Lyapunov functions derived from data to offer a control Lyapunov function (CLF) for the interconnected network, ensuring GAS certificate over the network. We demonstrate that while the computational complexity for designing a CLF increases polynomially with the network dimension using sum-of-squares (SOS) optimization, our compositional data-driven approach significantly mitigates it to \emph{linear} with respect to the number of subsystems. We showcase the efficacy of our data-driven approach over a set of benchmarks, involving physical networks with diverse interconnection topologies.

Paper Structure

This paper contains 18 sections, 4 theorems, 48 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem description
  3. Notation
  4. Individual Subsystems
  5. Interconnected Network
  6. GAS Property of Interconnected Network
  7. Data-driven construction of ISS Lyapunov functions
  8. Compositional construction of CLF for interconnected network
  9. Discussions
  10. Computational Complexity Analysis
  11. Limitations
  12. Simulation results
  13. Spacecraft Network: Fully-interconnected Topology
  14. Spacecraft Network: Ring Topology
  15. Academic Network: Binary Tree Topology
  16. ...and 3 more sections

Key Result

theorem 1

Consider an interconnected network $\Sigma=\mathcal{I}(\Sigma_1, \ldots,$$\Sigma_M)$, composed of $M$ subsystems $\Sigma_i$. Let there exist a control Lyapunov function (CLF) $\mathcal{V}\!:{\mathbb{R}}^n\to{\mathbb{R}}_0^+$, and constants $\underline{\alpha},\overline{\alpha},\kappa \in {\mathbb{R} where $x=[x_1;\dots;x_M]$ and $\mathsf{L} \mathcal{V}$ is the Lie derivative of $\mathcal{V}:{\math

Figures (6)

  • Figure 1: Different interconnection topologies employed in Table \ref{['tab:benchmark']}. In the binary tree topology shown in \ref{['fig:binary']}, $M=2^l-1$, where $l$ is the number of tree's levels.
  • Figure 2: Interconnected spacecraft network in a fully-interconnected topology: \ref{['fig:sc time ol']} and \ref{['fig:sc 3D ol']} depict the network's behavior without the controller, resulting in diverging trajectories. \ref{['fig:sc fully time']} illustrates the network's evolution under the synthesized controller, highlighting the states of an arbitrary subsystem in bold and shading those of the remaining subsystems. \ref{['fig:sc fully 3D']} visualizes trajectories of the network under the synthesized controller in 3D, demonstrating that all states converge to the origin.
  • Figure 3: Interconnected spacecraft network in a ring topology: \ref{['fig:sc ring time']} demonstrates the evolution of the network under the synthesized controller, highlighting the states of an arbitrary subsystem in bold and shading those of the remaining subsystems. \ref{['fig:sc ring 3D']} visualizes trajectories of the network under the synthesized controller in 3D, demonstrating that all states converge to the origin.
  • Figure 4: Interconnected academic network in a binary tree topology: \ref{['fig:academic time']} demonstrates the evolution of the network under the synthesized controller, highlighting the states of an arbitrary subsystem in bold and shading those of the remaining subsystems; \ref{['fig:CLF']} illustrates the ISS Lyapunov function $\mathcal{V}_i(x_i)$ of an arbitrary subsystem, demonstrating the fulfillment of condition \ref{['eq:ISS-con1']}.
  • Figure 5: Interconnected academic network in a star topology: \ref{['fig:academic star time']} demonstrates the evolution of the network under the synthesized controller, highlighting the states of an arbitrary subsystem in bold and shading those of the remaining subsystems; \ref{['fig:CLF star']} illustrates the ISS Lyapunov function $\mathcal{V}_i(x_i)$ of an arbitrary subsystem, demonstrating the fulfillment of condition \ref{['eq:ISS-con1']}.
  • ...and 1 more figures

Theorems & Definitions (14)

  • definition 1
  • definition 2
  • definition 3
  • theorem 1: sontag1996new
  • definition 4
  • Remark 1
  • lemma 1
  • Remark 2
  • Remark 3
  • theorem 2
  • ...and 4 more