Observability-blocking control using sparser and regional feedback for network synchronization processes

Abstract

The design of feedback control systems to block observability in a network synchronization model, i.e. to make the dynamics unobservable from measurements at a subset of the network’s nodes, is studied. First, a general design algorithm is presented for blocking observability at any group of $m$ nodes, by applying state feedback controls at $m+2$ specified actuation nodes. The algorithm is based on a method for eigenstructure assignment, which allows surgical modification of particular eigenvectors to block observability while preserving the remaining open-loop eigenstructure. Next, a sparser design is obtained, which leverages low-cardinality vertex cutsets separating the actuation and measurement locations in the network’s graph to reduce the required number of actuation nodes for observability blocking. Also, the design is modified to encompass regional feedback controls, which only use data from accessible nodes. The regional feedback design does not maintain the open-loop eigenstructure, but can be guaranteed to preserve stability via a time-scale argument. The sparser and regional feedback designs are illustrated with numerical examples.

Introduction

The controllability and observability of dynamical network processes with actuation/measurement at a subset of network nodes has been extensively researched during the last few years (Dhal and Roy, 2015, Li et al., 2020, Liu et al., 2011, Pasqualetti et al., 2014, Rahmani et al., 2009, Roy et al., 2016, Summers et al., 2015, Summers and Lygeros, 2014, Sundaram and Hadjicostis, 2012, Wang et al., 2016). These studies have largely focused on the relationship between a network’s native topology and controllability/observability. A key further question of interest is whether control systems in a network can be designed to facilitate or block observability/controllability from particular network locations, while preserving overall performance. Such design problems often arise when multiple control authorities or stakeholders have access to the network’s dynamics, and have cross-cutting goals in modulating the dynamics (whether cooperative or competitive). For instance, given increasing concern about cyber-attacks in the power grid, there is interest in designing wide-area control systems that not only damp oscillations but prevent adversaries from estimating or manipulating the dynamics (Sridhar, Hahn, & Govindarasu, 2011). Likewise, control initiatives in the air transportation system potentially could be designed to improve or prevent estimation of flow dynamics (Liu, Kwon, Aljanabi, & Hwang, 2012), At a different scale, controller designs for multi-vehicle systems may need to consider security from intruders that can probe a subset of vehicles (Xue, Wang, & Roy, 2014). Based on this motivation, here we examine the problem of designing controls at a subset of a network’s nodes, to make the network’s dynamics unobservable from measurements at remote locations.

Specifically, we focus in this study on a linear network synchronization model (Ren and Beard, 2005, Xiao and Boyd, 2004, Xiao et al., 2007) which can be actuated at a sparse set of network nodes. Our goal is to determine whether regional state feedback controls applied at these nodes can be used to enforce unobservability to monitoring at a set of remote nodes, while maintaining dynamical properties of the network. We study the design of such observability-blocking controllers, by applying and enhancing eigenstructure assignment methods in tandem with graph-theoretic and linear-algebraic techniques. The main contributions of the study are as follows:

• A general algorithm is obtained for designing state-feedback controllers which block observability at a set of measurement nodes through assignment of one or a few closed-loop eigenvectors (Algorithm 1). The algorithm is shown to yield control designs that block observability, while preserving the remaining open-loop eigenvectors and all the open-loop eigenvalues (Theorem 1, Theorem 2).

• A topology-exploiting scheme is developed to block observability which requires fewer actuation nodes relative to the general algorithm (Theorem 3). Specifically, it is shown that the required number of actuators depends on the minimum cardinality among vertex cutsets (i.e. sets of vertices whose removal partitions the graph) separating actuation and measurement locations in the network’s graph.

• The design scheme is further modified to accommodate regional feedback where only the states from a region or partition are used (Theorem 4). In this case, the observability-blocking control scheme modifies the eigenstructure of the open-loop system, but stability can still be guaranteed via a time-scale separation-based design (Theorem 5).

The developed methodology for observability-blocking controller design centrally draws on algebraic techniques for eigenstructure assignment, which originated in the classical works of Moore et al. (Klein and Moore, 1977, Moore, 1976) and others (Fahmy and O’reilly, 1983, Padula et al., 2021, Porter and D’azzo, 1978). These techniques have proved useful for a number of control applications, include e.g. aircraft control and malicious control of the power grid (DeMarco et al., 1996, Nieto-Wire and Sobel, 2014). Motivated by applications requiring only targeted eigenstructure assignment, we have recently adapted the classical eigenstructure assignment method for surgical eigenstructure assignment – i.e. placement of all eigenvalues and a subset of eigenvectors – subject to conditions only on the dictated eigenvectors (Al Maruf and Roy, 2021a, Maruf and Roy, 2020) (see also Lu, Chiang, and Thorp (1991)). The design algorithms presented here use and enhance the eigenstructure assignment methods introduced in Klein and Moore, 1977, Maruf and Roy, 2020, Moore, 1976.

The research described here also connects to a wide literature on controller design for built network processes, i.e. ones which have fixed topological interactions with sparse actuation and measurement capabilities, such as oscillation-damping in the power grid (Trudnowski, Smith, Short, & Pierre, 1991) and pinning control of synchronization processes (Yu, Chen, Lu, & Kurths, 2013). However, while this literature is mainly focused on modal properties of linearized networks models, our focus here is in shaping external or channel properties. Very recently, there has been a growing interest in shaping the input–output dynamics of networks (Besselink and Knorn, 2018, Li et al., 2020, Paridari et al., 2017, Roy et al., 2016, Stüdli et al., 2017); the research described here contributes to this effort.

The paper is organized as follows. The observability-blocking controller design problem is formulated in Section 2. In Section 3, the eigenstructure assignment methods which form the basis for our approach are briefly reviewed. The general algorithm for designing observability-blocking controllers is presented in Section 4, with graph-exploiting sparser and regional feedback methods developed in Sections 5 Sparser observability blocking using network graph cutsets, 6 Observability blocking using regional state feedback respectively. A brief conclusion is given in Section 7. Examples are presented throughout to illustrate the formal results. Proofs are relegated to an appendix to simplify the presentation. Some preliminary results related to observability-blocking control were presented in the conference paper Al Maruf and Roy (2019). Relative to Al Maruf and Roy (2019), the design algorithms are now presented in generality (e.g., encompassing the repeated eigenvalue case, general graph structures, etc.), an error in the general algorithm is corrected, the regional feedback design is developed, new examples are included, and the motivation for the work has been substantially updated. We also note that the dual problem of blocking controllability has been considered in the conference paper Al Maruf (2021b), however entirely different algorithms are obtained in this case.