arXiv subjects

Henrik Sandberg

Publications and source records attributed to Henrik Sandberg.

At least 37 records · Page 2Linked to original sources

arXiv2025

Fault Localisation in Infinite-Dimensional Linear Electrical Networks

We present a novel fault localisation methodology for linear time-invariant electrical networks with infinite-dimensional edge dynamics and uncertain fault dynamics. The theory accommodates instability and also bounded propagation delays in the network. The goal is to estimate the location of a fault along a given network edge, using sensors positioned arbitrarily throughout the network. Passive faults of unknown impedance are considered, along with stable faults of known impedance. To illustrate the approach, we tackle a significant use-case: a multi-conductor transmission line, with dynamics modelled by the Telegrapher's equation, subject to a line-to-ground fault. Frequency-domain insights are used to reformulate the general fault localisation problem into a non-convex scalar optimisation problem, of which the true fault location is guaranteed to be a global minimiser. Numerical experiments are run to quantify localisation performance over a range of fault resistances.

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arXiv2025

Feasibility of Randomized Detector Tuning for Attack Impact Mitigation

This paper considers the problem of detector tuning against false data injection attacks. In particular, we consider an adversary injecting false sensor data to maximize the state deviation of the plant, referred to as impact, whilst being stealthy. To minimize the impact of stealthy attacks, inspired by moving target defense, the operator randomly switches the detector thresholds. In this paper, we theoretically derive the sufficient (and in some cases necessary) conditions under which the impact of stealthy attacks can be made smaller with randomized switching of detector thresholds compared to static thresholds. We establish the conditions for the stateless ($χ^2$) and the stateful (CUSUM) detectors. The results are illustrated through numerical examples.

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arXiv2025

Anomaly Detection with LWE Encrypted Control

Detecting attacks using encrypted signals is challenging since encryption hides its information content. We present a novel mechanism for anomaly detection over Learning with Errors (LWE) encrypted signals without using decryption, secure channels, nor complex communication schemes. Instead, the detector exploits the homomorphic property of LWE encryption to perform hypothesis tests on transformations of the encrypted samples. The specific transformations are determined by solutions to a hard lattice-based minimization problem. While the test's sensitivity deteriorates with suboptimal solutions, similar to the exponential deterioration of the (related) test that breaks the cryptosystem, we show that the deterioration is polynomial for our test. This rate gap can be exploited to pick parameters that lead to somewhat weaker encryption but large gains in detection capability. Finally, we conclude the paper by presenting a numerical example that simulates anomaly detection, demonstrating the effectiveness of our method in identifying attacks.

cs.CR↗

arXiv2024

Modeling and Analysis of a Coupled SIS Bi-Virus Model

The paper deals with the setting where two viruses (say virus 1 and virus 2) coexist in a population, and they are not necessarily mutually exclusive, in the sense that infection due to one virus does not preclude the possibility of simultaneous infection due to the other. We develop a coupled bi-virus susceptible-infected-susceptible (SIS) model from a 4n-state Markov chain model, where n is the number of agents (i.e., individuals or subpopulation) in the population. We identify a sufficient condition for both viruses to eventually die out, and a sufficient condition for the existence, uniqueness and asymptotic stability of the endemic equilibrium of each virus. We establish a sufficient condition and multiple necessary conditions for local exponential convergence to the boundary equilibrium (i.e., one virus persists, the other one dies out) of each virus. Under mild assumptions on the healing rate, we show that there cannot exist a coexisting equilibrium where for each node there is a nonzero fraction infected only by virus 1; a nonzero fraction infected only by virus 2; but no fraction that is infected by both viruses 1 and 2. Likewise, assuming that healing rates are strictly positive, a coexisting equilibrium where for each node there is a nonzero fraction infected by both viruses 1 and 2, but no fraction is infected only by virus 1 (resp. virus 2) does not exist. Further, we provide a necessary condition for the existence of certain other kinds of coexisting equilibria. We show that, unlike the competitive bivirus model, the coupled bivirus model is not monotone. Finally, we illustrate our theoretical findings using an extensive set of in-depth simulations.

q-bio.PE↗

arXiv2024

Regret Lower Bounds for Learning Linear Quadratic Gaussian Systems

TWe establish regret lower bounds for adaptively controlling an unknown linear Gaussian system with quadratic costs. We combine ideas from experiment design, estimation theory and a perturbation bound of certain information matrices to derive regret lower bounds exhibiting scaling on the order of magnitude $\sqrt{T}$ in the time horizon $T$. Our bounds accurately capture the role of control-theoretic parameters and we are able to show that systems that are hard to control are also hard to learn to control; when instantiated to state feedback systems we recover the dimensional dependency of earlier work but with improved scaling with system-theoretic constants such as system costs and Gramians. Furthermore, we extend our results to a class of partially observed systems and demonstrate that systems with poor observability structure also are hard to learn to control.

cs.LG↗

arXiv2024

On well-posedness of the leak localization problem in parallel pipe networks

With the advent of integrated sensor technology (smart flow meters and pressure sensors), various new numerical algorithms for leak localization (a core element of water distribution system operation) have been developed. However, there is a lack of theory regarding the limitations of leak localization. In this work, we contribute to the development of such a theory by introducing an example water network structure with parallel pipes that is tractable for analytical treatment. We define the leak localization problem for this structure and show how many sensors and what conditions are needed for the well-posedness of the problem. We present a formula for the leak position as a function of measurements from these sensors. However, we also highlight the risk of finding false but plausible leak positions in the multiple pipes. We try to answer the questions of how and when the leaking pipe can be isolated. In particular, we show that nonlinearities in the pipes' head loss functions are essential for the well-posedness of the isolation problem. We propose procedures to get around the pitfall of multiple plausible leak positions.

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arXiv2023

Asymptotic Security using Bayesian Defense Mechanism with Application to Cyber Deception

This paper addresses the question whether model knowledge can guide a defender to appropriate decisions, or not, when an attacker intrudes into control systems. The model-based defense scheme considered in this study, namely Bayesian defense mechanism, chooses reasonable reactions through observation of the system's behavior using models of the system's stochastic dynamics, the vulnerability to be exploited, and the attacker's objective. On the other hand, rational attackers take deceptive strategies for misleading the defender into making inappropriate decisions. In this paper, their dynamic decision making is formulated as a stochastic signaling game. It is shown that the belief of the true scenario has a limit in a stochastic sense at an equilibrium based on martingale analysis. This fact implies that there are only two possible cases: the defender asymptotically detects the attack with a firm belief, or the attacker takes actions such that the system's behavior becomes nominal after a finite time step. Consequently, if different scenarios result in different stochastic behaviors, the Bayesian defense mechanism guarantees the system to be secure in an asymptotic manner provided that effective countermeasures are implemented. As an application of the finding, a defensive deception utilizing asymmetric recognition of vulnerabilities exploited by the attacker is analyzed. It is shown that the attacker possibly stops the attack even if the defender is unaware of the exploited vulnerabilities as long as the defender's unawareness is concealed by the defensive deception.

cs.CR↗

arXiv2023

Electrical Fault Localisation Over a Distributed Parameter Transmission Line

Motivated by the need to localise faults along electrical power lines, this paper adopts a frequency-domain approach to parameter estimation for an infinite-dimensional linear dynamical system with one spatial variable. Since the time of the fault is unknown, and voltages and currents are measured at only one end of the line, distance information must be extracted from the post-fault transients. To properly account for high-frequency transient behaviour, the line dynamics is modelled directly by the Telegrapher's equation, rather than the more commonly used lumped-parameter approximations. First, the governing equations are non-dimensionalised to avoid ill-conditioning. A closed-form expression for the transfer function is then derived. Finally, nonlinear least-squares optimisation is employed to search for the fault location. Requirements on fault bandwidth, sensor bandwidth and simulation time-step are also presented. The result is a novel end-to-end algorithm for data generation and fault localisation, the effectiveness of which is demonstrated via simulation.

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arXiv2023

Resilient Scheduling of Control Software Updates in Radial Power Distribution Systems

In response to newly found security vulnerabilities, or as part of a moving target defense, a fast and safe control software update scheme for networked control systems is highly desirable. We here develop such a scheme for intelligent electronic devices (IEDs) in power distribution systems, which is a solution to the so-called software update rollout problem. This problem seeks to minimize the makespan of the software rollout, while guaranteeing safety in voltage and current at all buses and lines despite possible worst-case update failure where malfunctioning IEDs may inject harmful amounts of power into the system. Based on the nonlinear DistFlow equations, we derive linear relations relating software update decisions to the worst-case voltages and currents, leading to a decision model both tractable and more accurate than previous models based on the popular linearized DistFlow equations. Under reasonable protection assumptions, the rollout problem can be formulated as a vector bin packing problem and instances can be built and solved using scalable computations. Using realistic benchmarks including one with 10,476 buses, we demonstrate that the proposed method can generate safe and effective rollout schedules in real-time.

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arXiv2023

Complexity reduction for resilient state estimation of uniformly observable nonlinear systems

A resilient state estimation scheme for uniformly observable nonlinear systems, based on a method for local identification of sensor attacks, is presented. The estimation problem is combinatorial in nature, and so many methods require substantial computational and storage resources as the number of sensors increases. To reduce the complexity, the proposed method performs the attack identification with local subsets of the measurements, not with the set of all measurements. A condition for nonlinear attack identification is introduced as a relaxed version of existing redundant observability condition. It is shown that an attack identification can be performed even when the state cannot be recovered from the measurements. As a result, although a portion of measurements are compromised, they can be locally identified and excluded from the state estimation, and thus the true state can be recovered. Simulation results demonstrate the effectiveness of the proposed scheme.

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arXiv2023

Strategic Monitoring of Networked Systems with Heterogeneous Security Levels

We consider a strategic network monitoring problem involving the operator of a networked system and an attacker. The operator aims to randomize the placement of multiple protected sensors to monitor and protect components that are vulnerable to attacks. We account for the heterogeneity in the components' security levels and formulate a large-scale maximin optimization problem. After analyzing its structure, we propose a three-step approach to approximately solve the problem. First, we solve a generalized covering set problem and run a combinatorial algorithm to compute an approximate solution. Then, we compute approximation bounds by solving a nonlinear set packing problem. To evaluate our solution approach, we implement two classical solution methods based on column generation and multiplicative weights updates, and test them on real-world water distribution and power systems. Our numerical analysis shows that our solution method outperforms the classical methods on large-scale networks, as it efficiently generates solutions that achieve a close to optimal performance and that are simple to implement in practice.

math.OC↗

arXiv2023

The Fundamental Limitations of Learning Linear-Quadratic Regulators

We present a local minimax lower bound on the excess cost of designing a linear-quadratic controller from offline data. The bound is valid for any offline exploration policy that consists of a stabilizing controller and an energy bounded exploratory input. The derivation leverages a relaxation of the minimax estimation problem to Bayesian estimation, and an application of Van Trees' inequality. We show that the bound aligns with system-theoretic intuition. In particular, we demonstrate that the lower bound increases when the optimal control objective value increases. We also show that the lower bound increases when the system is poorly excitable, as characterized by the spectrum of the controllability gramian of the system mapping the noise to the state and the $\mathcal{H}_\infty$ norm of the system mapping the input to the state. We further show that for some classes of systems, the lower bound may be exponential in the state dimension, demonstrating exponential sample complexity for learning the linear-quadratic regulator offline.

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arXiv2023

Privacy Preserving Set-Based Estimation Using Partially Homomorphic Encryption

The set-based estimation has gained a lot of attention due to its ability to guarantee state enclosures for safety-critical systems. However, collecting measurements from distributed sensors often requires outsourcing the set-based operations to an aggregator node, raising many privacy concerns. To address this problem, we present set-based estimation protocols using partially homomorphic encryption that preserve the privacy of the measurements and sets bounding the estimates. We consider a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Sets are represented by zonotopes and constrained zonotopes as they can compactly represent high-dimensional sets and are closed under linear maps and Minkowski addition. By selectively encrypting parameters of the set representations, we establish the notion of encrypted sets and intersect sets in the encrypted domain, which enables guaranteed state estimation while ensuring privacy. In particular, we show that our protocols achieve computational privacy using the cryptographic notion of computational indistinguishability. We demonstrate the efficiency of our approach by localizing a real mobile quadcopter using ultra-wideband wireless devices.

cs.CR↗

arXiv2023

Attack Impact Evaluation for Stochastic Control Systems through Alarm Flag State Augmentation

This note addresses the problem of evaluating the impact of an attack on discrete-time nonlinear stochastic control systems. The problem is formulated as an optimal control problem with a joint chance constraint that forces the adversary to avoid detection throughout a given time period. Due to the joint constraint, the optimal control policy depends not only on the current state, but also on the entire history, leading to an explosion of the search space and making the problem generally intractable. However, we discover that the current state and whether an alarm has been triggered, or not, is sufficient for specifying the optimal decision at each time step. This information, which we refer to as the alarm flag, can be added to the state space to create an equivalent optimal control problem that can be solved with existing numerical approaches using a Markov policy. Additionally, we note that the formulation results in a policy that does not avoid detection once an alarm has been triggered. We extend the formulation to handle multi-alarm avoidance policies for more reasonable attack impact evaluations, and show that the idea of augmenting the state space with an alarm flag is valid in this extended formulation as well.

math.OC↗

arXiv2023

Scale Fragilities in Localized Consensus Dynamics

We consider distributed consensus in networks where the agents have integrator dynamics of order two or higher ($n\ge 2$). We assume all feedback to be localized in the sense that each agent has a bounded number of neighbors and consider a scaling of the network through the addition of agents in a modular manner, i.e., without re-tuning controller gains upon addition. We show that standard consensus algorithms, which rely on relative state feedback, are subject to what we term scale fragilities, meaning that stability is lost as the network scales. For high-order agents ($n\ge 3$), we prove that no consensus algorithm with fixed gains can achieve consensus in networks of any size. That is, while a given algorithm may allow a small network to converge, it causes instability if the network grows beyond a certain finite size. This holds in families of network graphs whose algebraic connectivity, that is, the smallest non-zero Laplacian eigenvalue, is decreasing towards zero in network size (e.g. all planar graphs). For second-order consensus ($n = 2$) we prove that the same scale fragility applies to directed graphs that have a complex Laplacian eigenvalue approaching the origin (e.g. directed ring graphs). The proofs for both results rely on Routh-Hurwitz criteria for complex-valued polynomials and hold true for general directed network graphs. We survey classes of graphs subject to these scale fragilities, discuss their scaling constants, and finally prove that a sub-linear scaling of nodal neighborhoods can suffice to overcome the issue.

math.OC↗

arXiv2022

Comparison of encrypted control approaches and tutorial on dynamic systems using LWE-based homomorphic encryption

Encrypted control has been introduced to protect controller data by encryption at the stage of computation and communication, by performing the computation directly on encrypted data. In this article, we first review and categorize recent relevant studies on encrypted control. Approaches based on homomorphic encryption, multi-party computation, and secret sharing are introduced, compared, and then discussed with respect to computational complexity, communication load, enabled operations, security, and research directions. We proceed to discuss a current challenge in the application of homomorphic encryption to dynamic systems, where arithmetic operations other than integer addition and multiplication are limited. We also introduce a homomorphic cryptosystem called ``GSW-LWE'' and discuss its benefits that allow for recursive multiplication of encrypted dynamic systems, without use of computationally expensive bootstrapping techniques.

cs.CR↗

arXiv2022

How are policy gradient methods affected by the limits of control?

We study stochastic policy gradient methods from the perspective of control-theoretic limitations. Our main result is that ill-conditioned linear systems in the sense of Doyle inevitably lead to noisy gradient estimates. We also give an example of a class of stable systems in which policy gradient methods suffer from the curse of dimensionality. Our results apply to both state feedback and partially observed systems.

math.OC↗

arXiv2022

A Bayesian Nash equilibrium-based moving target defense against stealthy sensor attacks

We present a moving target defense strategy to reduce the impact of stealthy sensor attacks on feedback systems. The defender periodically and randomly switches between thresholds from a discrete set to increase the uncertainty for the attacker and make stealthy attacks detectable. However, the defender does not know the exact goal of the attacker but only the prior of the possible attacker goals. Here, we model one period with a constant threshold as a Bayesian game and use the Bayesian Nash equilibrium concept to find the distribution for the choice of the threshold in that period, which takes the defender's uncertainty about the attacker into account. To obtain the equilibrium distribution, the defender minimizes its cost consisting of the cost for false alarms and the cost induced by the attack. We present a necessary and sufficient condition for the existence of a moving target defense and formulate a linear program to determine the moving target defense. Furthermore, we present a closed-form solution for the special case when the defender knows the attacker's goals. The results are numerically evaluated on a four-tank process.

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