Observability-Blocking Controls for Double-Integrator and Higher Order Integrator Networks

TL;DR

This work addresses blocking observability in double-integrator and higher-order integrator networks by designing state-feedback controllers that render chosen measurements unobservable while preserving the open-loop eigenstructure. It develops an DIN-specific eigenstructure-assignment algorithm requiring $q\ge m+2$ actuators, and further achieves sparser designs by exploiting network cutsets, with a condition$\lambda_p^2$ not an eigenvalue of a reduced Laplacian $\mathbf{L}_g$. The results generalize to $N$-th order integrator networks, including a sparse cutset-based approach (requiring $q\ge |\mathcal{V}_{cut}|+2$) and corresponding lemmas/theorems. A numerical example on an 11-node network demonstrates effective observability blocking and validates the theoretical claims, highlighting practical implications for privacy and security in multi-agent systems.

Abstract

The design of state-feedback controls to block observability at remote nodes is studied for double integrator network (DIN) and higher order integrator network models. A preliminary design algorithm is presented first for DIN that requires $m+2$ actuation nodes to block observability for the measurement obtained from a set of $m$ nodes. The algorithm is based on eigenstructure assignment technique and leverages the properties of the eigenvectors in DIN. Next, the topological structure of the network is exploited to reduce the number of controllers required for blocking observability. The number of actuation nodes in sparser design depends on the cardinality of a cutset separating the actuation and measurement locations. Later, the design principles are generalized for blocking observability in $N$-th order integrator network models.

Figures (2)

• Figure 1: Graph $\mathcal{G}$ of the network with node-cutset between actuation and measurement nodes.
• Figure 2: Network graph with double-integrator dynamics used in the numerical example