Global network control from local information

TL;DR

The paper addresses scalable control of large networks when full global state information cannot be reconstructed in real time. It introduces a non-iterative, closed-form feedforward control law that relies on local state information within state information neighborhoods (SINs) and avoids explicit inversion of the full controllability Gramian $W_f$. A sparse inverse $X$ of $W_f$ is obtained via a localization-based least-squares approximation, yielding SINs whose size is governed by the condition number $\kappa$ of $W_f$ and a tunable parameter $q$. Results on Erdős-Rényi, random geometric, and lattice networks, plus an Iceland power grid case, show that well-conditioned Gramians permit very small SINs with minimal performance loss, enabling efficient control of large networks with limited information exchange. The work also suggests extensions to nonlinear systems through feedforward model predictive control, positioning the approach as a building block for distributed control and estimation in large-scale networks.

Abstract

In the classical control of network systems, the control actions on a node are determined as a function of the states of all nodes in the network. Motivated by applications where the global state cannot be reconstructed in real time due to limitations in the collection, communication, and processing of data, here we introduce a control approach in which the control actions can be computed as a function of the states of the nodes within a limited state information neighborhood. The trade-off between the control performance and the size of this neighborhood is primarily determined by the condition number of the controllability Gramian. Our theoretical results are supported by simulations on regular and random networks and are further illustrated by an application to the control of power-grid synchronization. We demonstrate that for well-conditioned Gramians, there is no significant loss of control performance as the size of the state information neighborhood is reduced, allowing efficient control of large networks using only local information.

Figures (4)

• Figure 1: Ordered magnitudes of the entries of $W_f^{-1}$ for (a) an ERN ($N=598$), (b) an RGN ($N=702$), and (c) a lattice ($N=35^2$), for $B=B_1$ (black), $B_2$ (blue), and $B_3$ (red). Inset in (c): corresponding envelope of the off-diagonal decay for the middle row as a function of the column. (d) SIN and effective SINs for the matrix $\mathcal{Q}$ around a central node for the $B=B_{1}$ case in (c) and thresholds $\delta_{1}=0$, $\delta_{2}=10^{-6}$, and $\delta_{3}=10^{-3}$. Other parameters: $f=5$ in (a-d) and $q=3$ in (d) param. Details on network generation, sparsity of $W_{f}$, localization of $W_{f}^{-1}$, and (effective) SINs are presented in Supplementary Material, Secs. S3 and S4.
• Figure 2: Control error for (a, b) ERNs and (c, d) RGNs as a function of (a, c) $q$ and (b, d) $N$, where $B=B_1$ ($\bullet$), $B_2$ ($\blacklozenge$), and $B_3$ ($\boldsymbol{*}$). The fixed parameters are $f=3$ in (a-d), $N=598$ in (a), $N=702$ in (c), and $q=2$ in (b, d) param. Each data point is an average over $100$ realizations of matrix $A$ in (a, c) and of network topologies in (b, d). For the sparsity patterns of the associated matrices $\mathcal{Q}$ and $\mathcal{R}$, see Supplementary Material, Sec. S4.3.
• Figure 3: Control error versus the fraction of controlled nodes in (a) an ERN ($N=598$) and (b) an RGN ($N=702$), for $q=1$ ($\blacksquare$), $q=2$ ($\bullet$), and $q=3$ ($\blacktriangle$). Each point is an average over $100$ realizations of randomly chosen nodes to control. Insets: corresponding condition numbers of $W_f$ color-coded as in the main panels. The other settings are $f=5$ and $B_{ii} = B_{ii}^{(1)}$ if node $i$ is controlled (and zero otherwise) param.
• Figure 4: Control of Iceland's power grid when driving the generators to the desired synchronous state. (a) Control error for several $f$ as a function of $q$. Each point is an average over $100$ realizations of $\mathbf{x}(0)$ generated by randomly perturbing each state variable by $\pm 0.01$ away from $\mathbf{x}^d$, and $B$ is the identity matrix. (b) SIN of the node marked by the arrow for the matrix $\mathcal{Q}$ and $f=q=2$. Details of the dynamics are presented in Supplementary Material, Sec. S9.