Data-Driven Control of Large-Scale Networks with Formal Guarantees: A Small-Gain Free Approach

Abstract

This paper offers a data-driven divide-and-conquer strategy to analyze large-scale interconnected networks, characterized by both unknown mathematical models and interconnection topologies. Our data-driven scheme treats an unknown network as an interconnection of individual agents (a.k.a. subsystems) and aims at constructing their symbolic models, referred to as discrete-domain representations of unknown agents, by collecting data from their trajectories. The primary objective is to synthesize a control strategy that guarantees desired behaviors over an unknown network by employing local controllers, derived from symbolic models of individual agents. To achieve this, we leverage the concept of alternating sub-bisimulation function (ASBF) to capture the closeness between state trajectories of each unknown agent and its data-driven symbolic model. Under a newly developed data-driven compositional condition, we then establish an alternating bisimulation function (ABF) between an unknown network and its symbolic model, based on ASBFs of individual agents, while providing correctness guarantees. Despite the sample complexity in existing work being exponential with respect to the network size, we demonstrate that our divide-and-conquer strategy significantly reduces it to a linear scale with respect to the number of agents. We also showcase that our data-driven compositional condition does not necessitate the traditional small-gain condition, which demands precise knowledge of the interconnection topology for its fulfillment. We apply our data-driven findings to three benchmarks comprising unknown networks with an arbitrary, a-priori undefined number of agents and unknown interconnection topologies.

Figures (6)

• Figure 1: Illustration of the procedure for constructing the symbolic model. The center of each cell is taken as a representative point, corresponding to an element of $\hat{X}$.
• Figure 2: Data-driven ASBF is non-negative in the whole range of state space (a), while satisfying condition \ref{['Eq:8_21']} (b).
• Figure 3: Closed-loop state trajectories of three arbitrary rooms by designing controllers via their data-driven symbolic models. While two out of the three initial conditions are positioned at the boundaries of the safe set, the designed controllers perfectly maintain the trajectories within the safe set.
• Figure 4: In the first scenario, we assume that each vehicle's trajectories start from a specific initial set . As shown in Figure \ref{['subfig2']}, even though this initial set shares a border with the unsafe set , all trajectories remain within the safe set and never enter the unsafe set. In the second scenario, we assume that trajectories of vehicles can start anywhere inside the safe set. As illustrated in Figure \ref{['subfig3']}, all trajectories never leave the safe set under any circumstances. While only $10$ vehicles are selected for demonstration purposes, we observed that safety is maintained for all $M$ vehicles.
• Figure 5: State trajectories and corresponding control inputs of four representative subsystems, obtained through controllers synthesized via data-driven symbolic models.

Theorems & Definitions (26)

• Definition 1
• Remark 1
• Definition 2
• Remark 2
• Definition 3
• Definition 4
• Remark 3
• Definition 5
• Remark 4
• Definition 6