From Data to Sliding Mode Control of Uncertain Large-Scale Networks with Unknown Dynamics

TL;DR

The paper tackles stabilization of large-scale networks with unknown subprocess dynamics by developing a data-driven, compositional approach. It uses two input-state trajectories per nominal subsystem to synthesize $ISS$ Lyapunov functions and local controllers via data-driven SDP formulations, and designs local ISM controllers to reject matched disturbances. A small-gain based compositional framework then yields a network CLF that guarantees global asymptotic stability for the nominal network, extended to perturbed networks. The method is validated on five topologies (fully connected, ring, binary tree, star, line), demonstrating GAS under external perturbations and highlighting scalability advantages. This work advances practical stability guarantees for uncertain, interconnected systems using limited data and formal compositional analysis.

Abstract

Large-scale interconnected networks, composed of multiple low-dimensional subsystems, serve as a crucial framework for modeling a wide range of real-world applications. Despite offering computational scalability, the inherent interdependence among subsystems poses significant challenges to the effective control of such networks. This complexity is further exacerbated in the presence of external perturbations and when the dynamics of individual subsystems, and accordingly the overall network, are unknown-scenarios frequently encountered in modern practical applications. In this paper, we develop a compositional data-driven approach to ensure the global asymptotic stability (GAS) of large-scale nonlinear networks with unknown mathematical models, subjected to external perturbations. To achieve this, we first gather two sets of data from each unknown nominal subsystem without perturbation, which we refer to as two input-state trajectories. The collected data from each subsystem is then utilized to design an input-to-state stable (ISS) Lyapunov function and its corresponding controller for each nominal subsystem, rendering them ISS. To cancel the effect of external perturbations on the dynamic of each subsystem, and accordingly the whole network, we then design a local integral sliding mode (ISM) controller for each subsystem using the collected data. Under a small-gain compositional condition, we employ data-driven ISS Lyapunov functions designed for subsystems and construct a control Lyapunov function for the network, rendering the assurance of GAS property over the nominal network. We then extend this compositional result to network perturbed models, demonstrating that the synthesized ISM controllers ensure the GAS property even in the presence of perturbations.

Figures (5)

• Figure 1: The distinct interconnection topologies used in the simulations (cf. Table \ref{['table: info']}). We note that in the binary tree topology, the number of subsystems is computed as $N = 2^l - 1$, where $l$ denotes the number of tree levels.
• Figure 2: $10$ representative trajectories of the network (with the fully connected topology) with the designed local ISS controllers $u_i^\ast$, starting from different initial conditions $x(0) \in [-1, \, 1]$, illustrating fluctuations around the origin.
• Figure 3: $10$ representative trajectories of the network (with the fully connected topology) under the designed local controllers as in Corollary \ref{['Corollary: ext']}, starting from different initial conditions in $x(0) \in [-5\times 10^5, \, 5\times 10^5].$
• Figure 4: Representative behavior of the sliding variable components. For clarity, only the sliding variable components of a single subsystem are shown.
• Figure 5: Figure \ref{['fig:subfig1_1']} illustrates $10$ representative trajectories of the network (with a line topology) governed by the locally designed controllers, as described in Corollary \ref{['Corollary: ext']}, with initial states $x(0) \in [-5000, 5000]$. Moreover, Figure \ref{['fig:subfig2_1']} depicts the sample behavior of the sliding variable components.

Theorems & Definitions (21)

• Definition 1: ctia-UNCS
• Remark 1: Dictionary $\mathcal{Z}_i(x_i)$
• Definition 2: Interconnected Network
• Definition 3: GAS Property
• Theorem 1: CLFs for Interconnected Networks
• Definition 4: ISS Lyapunov Functions
• Remark 2: On Samples \ref{['eq: Sip']}
• Lemma 1: Data-based Representation of $\Upsilon_i^\ast$
• proof
• Remark 3: Richness of Data