How to synchronize a network of systems without knowing their model?

Imagine a squad of robots that must move in perfect harmony, but you know nothing about how each robot works. Traditionally, engineers use a precise mathematical model of every unit to synchronize them. If the model is unknown, the classic method fails.

A new study proposes a data‑driven solution: collect input‑state data from a single robot, then, using the network’s connection map (the topology), directly compute from the data a feedback gain that makes all robots move in sync.

The trick has two steps. First, a mathematical condition derived from the data tells whether the network is "synchronizable." If the condition holds, you can build the control gain without ever writing the robot’s equations. Second, that gain is inserted into the network’s communication protocol, so each robot adjusts its motion based on its neighbors.

The authors tested the method in a simulation: they took data from one linear system, calculated the gain, and watched all network nodes converge to the same state. The result shows that even without knowing the internal model, it’s possible to make a network of systems "dance" together. This idea opens the door to more flexible control in robotics, smart grids, and IoT systems, where full models are often hard to obtain.

The work relies on homogeneous linear time‑invariant (LTI) systems interconnected through a consensus protocol. The network topology is described by a Laplacian matrix L, whose structure influences the eigenvalues of the overall system. Traditionally, synchronization requires solving a Riccati equation or designing a gain K that stabilizes (A‑BK)L, where A and B are the state and input matrices of the LTI model.

In the data‑driven context, we assume access to a sequence of data {u(k), x(k)} collected from a single node, with u(k) the input and x(k) the state at discrete instants k. We build the data matrices X = [x(0) … x(T‑1)] and U = [u(0) … u(T‑1)], as well as the successor state matrix X⁺ = [x(1) … x(T)]. Persistence of excitation guarantees that the pair (U, X) is information‑rich.

Synchronizability conditions are derived by requiring the existence of a matrix K such that (A‑BK) is stable on all subspaces orthogonal to the consensus vector 1ₙ. Using Willems' fundamental lemma, we obtain a parametric description of all (A,B) compatible with the data: [A B] = X⁺V + N, where V is a pseudoinverse matrix and N belongs to the kernel of [X;U]. Substituting this parametrization into the stability condition yields Linear Matrix Inequalities (LMIs) that depend only on the data. Feasibility of these LMIs is both necessary and sufficient for synchronization.

Once feasibility is confirmed, the gain K can be extracted by solving a convex optimization problem, for instance minimizing the H₂ norm of the closed‑loop system. The result is an explicit K computed entirely from data, guaranteeing that all nodes converge to consensus with a decay rate dictated by the eigenvalues of (A‑BK)L.

The paper also demonstrates robustness to measurement noise by introducing tolerance margins in the LMIs. The numerical simulation uses a fourth‑order system with a complete‑graph topology, showing that the data‑driven gain achieves performance comparable to a model‑based K, without any prior knowledge of the system model.