arXiv subjects

Henrik Sandberg

Publications and source records attributed to Henrik Sandberg.

At least 55 records · Page 3Linked to original sources

arXiv2022

Finite sample guarantees for quantile estimation: An application to detector threshold tuning

In threshold-based anomaly detection, we want to tune the threshold of a detector to achieve an acceptable false alarm rate. However, tuning the threshold is often a non-trivial task due to unknown detector output distributions. A detector threshold that provides an acceptable false alarm rate is equivalent to a specific quantile of the detector output distribution. Therefore, we use quantile estimators based on order statistics to estimate the detector threshold. The estimation of quantiles from sample data has a more than a century long tradition and we provide three different distribution-free finite sample guarantees for a class of quantile estimators. The first is based on the Dworetzky-Kiefer-Wolfowitz inequality, the second utilizes the Vysochanskij-Petunin inequality, and the third is based on exact confidence intervals for a beta distribution. These guarantees are then compared and used in the detector threshold tuning problem. We use both simulated data as well as data obtained from an experimental setup with the Temperature Control Lab to validate the guarantees provided.

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arXiv2022

Experimental evaluation of sensor attacks and defense mechanisms in feedback systems

In this work, we evaluate theoretical results on the feasibility of, the worst-case impact of, and defense mechanisms against a stealthy sensor attack in an experimental setup. We demonstrate that for a controller with stable dynamics the stealthy sensor attack is possible to conduct and the theoretical worst-case impact is close to the achieved practical one. However, although the attack should theoretically be possible when the controller has integral action, we show that the integral action slows the attacker down and the attacker is not able to remain stealthy if it has not perfect knowledge of the controller state. In addition to that, we investigate the effect of different anomaly detectors on the attack impact and conclude that the impact under detectors with internal dynamics is smaller. Finally, we use noise injection into the controller dynamics to unveil the otherwise stealthy attacks.

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arXiv2022

Privacy Guarantees for Cloud-based State Estimation using Partially Homomorphic Encryption

The privacy aspect of state estimation algorithms has been drawing high research attention due to the necessity for a trustworthy private environment in cyber-physical systems. These systems usually engage cloud-computing platforms to aggregate essential information from spatially distributed nodes and produce desired estimates. The exchange of sensitive data among semi-honest parties raises privacy concerns, especially when there are coalitions between parties. We propose two privacy-preserving protocols using Kalman filter and partially homomorphic encryption of the measurements and estimates while exposing the covariances and other model parameters. We prove that the proposed protocols achieve satisfying computational privacy guarantees against various coalitions based on formal cryptographic definitions of indistinguishability. We evaluate the proposed protocols to demonstrate their efficiency using data from a real testbed.

cs.CR↗

arXiv2022

Green Routing Game: Strategic Logistical Planning using Mixed Fleets of ICEVs and EVs

This paper introduces a "green" routing game between multiple logistic operators (players), each owning a mixed fleet of internal combustion engine vehicle (ICEV) and electric vehicle (EV) trucks. Each player faces the cost of delayed delivery (due to charging requirements of EVs) and a pollution cost levied on the ICEVs. This cost structure models: 1) limited battery capacity of EVs and their charging requirement; 2) shared nature of charging facilities; 3) pollution cost levied by regulatory agency on the use of ICEVs. We characterize Nash equilibria of this game and derive a condition for its uniqueness. We also use the gradient projection method to compute this equilibrium in a distributed manner. Our equilibrium analysis is useful to analyze the trade-off faced by players in incurring higher delay due to congestion at charging locations when the share of EVs increases versus a higher pollution cost when the share of ICEVs increases. A numerical example suggests that to increase marginal pollution cost can dramatically reduce inefficiency of equilibria.

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arXiv2022

Attack Impact Evaluation by Exact Convexification through State Space Augmentation

We address the attack impact evaluation problem for control system security. We formulate the problem as a Markov decision process with a temporally joint chance constraint that forces the adversary to avoid being detected throughout the considered time period. Owing to the joint constraint, the optimal control policy depends not only on the current state but also on the entire history, which leads to the explosion of the search space and makes the problem generally intractable. It is shown that whether an alarm has been triggered or not, in addition to the current state is sufficient for specifying the optimal decision at each time step. Augmentation of the information to the state space induces an equivalent convex optimization problem, which is tractable using standard solvers.

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arXiv2022

Leakage Localization in Water Distribution Networks: A Model-Based Approach

The paper studies the problem of leakage localization in water distribution networks. For the case of a single pipe that suffers from a single leak, by taking recourse to pressure and flow measurements, and assuming those are noiseless, we provide a closed-form expression for leak localization, leak exponent and leak constant. For the aforementioned setting, but with noisy pressure and flow measurements, an expression for estimating the location of the leak is provided. Finally, assuming the existence of a single leak, for a network comprising of more than one pipe and assuming that the network has a tree structure, we provide a systematic procedure for determining the leak location, the leak exponent, and the leak constant

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arXiv2022

Data-Driven Set-Based Estimation using Matrix Zonotopes with Set Containment Guarantees

We propose a method to perform set-based state estimation of an unknown dynamical linear system using a data-driven set propagation function. Our method comes with set-containment guarantees, making it applicable to safety-critical systems. The method consists of two phases: (1) an offline learning phase where we collect noisy input-output data to determine a function to propagate the state-set ahead in time; and (2) an online estimation phase consisting of a time update and a measurement update. It is assumed that known finite sets bound measurement noise and disturbances, but we assume no knowledge of their statistical properties. These sets are described using zonotopes, allowing efficient propagation and intersection operations. We propose a new approach to compute a set of models consistent with the data and noise-bound, given input-output data in the offline phase. The set of models is utilized in replacing the unknown dynamics in the data-driven set propagation function in the online phase. Then, we propose two approaches to perform the measurement update. Simulations show that the proposed estimator yields state sets comparable in volume to the 3σ confidence bounds obtained by a Kalman filter approach, but with the addition of state set-containment guarantees. We observe that using constrained zonotopes yields smaller sets but with higher computational costs than unconstrained ones.

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arXiv2022

Single Trajectory Nonparametric Learning of Nonlinear Dynamics

Given a single trajectory of a dynamical system, we analyze the performance of the nonparametric least squares estimator (LSE). More precisely, we give nonasymptotic expected $l^2$-distance bounds between the LSE and the true regression function, where expectation is evaluated on a fresh, counterfactual, trajectory. We leverage recently developed information-theoretic methods to establish the optimality of the LSE for nonparametric hypotheses classes in terms of supremum norm metric entropy and a subgaussian parameter. Next, we relate this subgaussian parameter to the stability of the underlying process using notions from dynamical systems theory. When combined, these developments lead to rate-optimal error bounds that scale as $T^{-1/(2+q)}$ for suitably stable processes and hypothesis classes with metric entropy growth of order $δ^{-q}$. Here, $T$ is the length of the observed trajectory, $δ\in \mathbb{R}_+$ is the packing granularity and $q\in (0,2)$ is a complexity term. Finally, we specialize our results to a number of scenarios of practical interest, such as Lipschitz dynamics, generalized linear models, and dynamics described by functions in certain classes of Reproducing Kernel Hilbert Spaces (RKHS).

cs.LG↗

arXiv2021

On the confidentiality of controller states under sensor attacks

With the emergence of cyber-attacks on control systems it has become clear that improving the security of control systems is an important task in today's society. We investigate how an attacker that has access to the measurements transmitted from the plant to the controller can perfectly estimate the internal state of the controller. This attack on sensitive information of the control loop is, on the one hand, a violation of the privacy, and, on the other hand, a violation of the security of the closed-loop system if the obtained estimate is used in a larger attack scheme. Current literature on sensor attacks often assumes that the attacker has already access to the controller's state. However, this is not always possible. We derive conditions for when the attacker is able to perfectly estimate the controller's state. These conditions show that if the controller has unstable poles a perfect estimate of the controller state is not possible. Moreover, we propose a defence mechanism to render the attack infeasible. This defence is based on adding uncertainty to the controller dynamics. We also discuss why an unstable controller is only a good defence for certain plants. Finally, simulations with a three-tank system verify our results.

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arXiv2021

Disconnection-aware Attack Detection and Isolation with Separation-based Detector Reconfiguration

This study addresses incident handling during an adverse event for dynamical networked control systems. Incident handling can be divided into five steps: detection, analysis, containment, eradication, and recovery. For networked control systems, the containment step can be conducted through physical disconnection of an attacked subsystem. In accordance with the disconnection, the equipped attack detection unit should be reconfigured to maintain its detection capability. In particular, separating the detection subunit associated with the disconnected subsystem is considered as a specific reconfiguration scheme in this study. This paper poses the problem of disconnection-aware attack detection and isolation with the separation-based detector reconfiguration. The objective is to find an attack detection unit that preserves its detection and isolation capability even under any possible disconnection and separation. The difficulty arises from network topology variation caused by disconnection that can possibly lead to stability loss of the distributed observer inside the attack detection unit. A solution is proposed based on an existing controller design technique referred to as retrofit control. Furthermore, an application to low-voltage power distribution networks with distributed generation is exhibited. Numerical examples evidence the practical use of the proposed method through a benchmark distribution network.

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arXiv2021

Multi-Layer SIS Model with an Infrastructure Network

This paper deals with the spread of diseases over both a population network and an infrastructure network. We develop a layered networked spread model for a susceptible-infected-susceptible (SIS) pathogen-borne disease spreading over a human contact network and an infrastructure network, and refer to it as a layered networked susceptible-infected-water-susceptible (SIWS) model. The SIWS network is in the healthy state (also referred to as the disease-free equilibrium) if none of the individuals in the population are infected nor is the infrastructure network contaminated; otherwise, we say that the network is in the endemic state (also referred to as the endemic equilibrium). First, we establish sufficient conditions for local exponential stability and global asymptotic stability (GAS) of the healthy state. Second, we provide sufficient conditions for existence, uniqueness, and GAS of the endemic state. Building off of these results, we provide a necessary, and sufficient, condition for the healthy state to be the unique equilibrium of our model. Third, we show that the endemic equilibrium of the SIWS model is worse than that of the networked SIS model without any infrastructure network, in the sense that at least one subpopulation has strictly larger infection proportion at the endemic equilibrium in the former model than that in the latter. Fourth, we study an observability problem, and, assuming that the measurements of the sickness-levels of the human contact network are available, provide a necessary and sufficient condition for estimation of the pathogen levels in the infrastructure network. Furthermore, we provide another sufficient, but not necessary, condition for estimation of pathogen levels in the infrastructure network.

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arXiv2021

How to Secure Distributed Filters Under Sensor Attacks

We study how to secure distributed filters for linear time-invariant systems with bounded noise under false-data injection attacks. A malicious attacker is able to arbitrarily manipulate the observations for a time-varying and unknown subset of the sensors. We first propose a recursive distributed filter consisting of two steps at each update. The first step employs a saturation-like scheme, which gives a small gain if the innovation is large corresponding to a potential attack. The second step is a consensus operation of state estimates among neighboring sensors. We prove the estimation error is upper bounded if the filter parameters satisfy a condition. We further analyze the feasibility of the condition and connect it to sparse observability in the centralized case. When the attacked sensor set is known to be time-invariant, the secured filter is modified by adding an online local attack detector. The detector is able to identify the attacked sensors whose observation innovations are larger than the detection thresholds. Also, with more attacked sensors being detected, the thresholds will adaptively adjust to reduce the space of the stealthy attack signals. The resilience of the secured filter with detection is verified by an explicit relationship between the upper bound of the estimation error and the number of detected attacked sensors. Moreover, for the noise-free case, we prove that the state estimate of each sensor asymptotically converges to the system state under certain conditions. Numerical simulations are provided to illustrate the developed results.

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arXiv2021

A Model Randomization Approach to Statistical Parameter Privacy

In this paper, we study a privacy filter design problem for a sequence of sensor measurements whose joint probability density function (p.d.f.) depends on a private parameter. To ensure parameter privacy, we propose a filter design framework which consists of two components: a randomizer and a nonlinear transformation. The randomizer takes the private parameter as input and randomly generates a pseudo parameter. The nonlinear mapping transforms the measurements such that the joint p.d.f. of the filter's output depends on the pseudo parameter rather than the private parameter. It also ensures that the joint p.d.f. of the filter's output belongs to the same family of distributions as that of the measurements. The nonlinear transformation has a feedforward-feedback structure that allows real-time and causal generation of the disguised measurements with low complexity using a recursive structure. The design of the randomizer is formulated as an optimization problem subject to a privacy constraint, in terms of mutual information, and it is shown that the optimal randomizer is the solution of a convex optimization problem. Using information-theoretic inequalities, we show that the performance of any estimator of the private parameter, based on the output of the privacy filter, is limited by the privacy constraint. The structure of the nonlinear transformation is studied in the special cases of independent and identically distributed, Markovian, and Gauss-Markov measurements. Our results show that the privacy filter in the Gauss-Markov case can be implemented as two one-step ahead Kalman predictors and a set of minimum mean square error predictors. The Kalman predictors significantly reduce the complexity of computing the disguised measurements. A numerical example on occupancy privacy in a building automation system illustrates the approach.

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arXiv2021

Epistemic Signaling Games for Cyber Deception with Asymmetric Recognition

This study provides a model of cyber deception with asymmetric recognition represented by private beliefs. Signaling games, which are often used in existing works, are built on the implicit premise that the receiver's belief is public information. However, this assumption, which leads to symmetric recognition, is unrealistic in adversarial decision making. For a precise evaluation of risks arising from cognitive gaps, this paper proposes epistemic signaling games based on the Mertens-Zamir model, which explicitly quantifies players' asymmetric recognition. Equilibria of the games are analytically characterized with an interpretation.

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arXiv2021

Geometrical Characterization of Sensor Placement for Cone-Invariant and Multi-Agent Systems against Undetectable Zero-Dynamics Attacks

Undetectable attacks are an important class of malicious attacks threatening the security of cyber-physical systems, which can modify a system's state but leave the system output measurements unaffected, and hence cannot be detected from the output. This paper studies undetectable attacks on cone-invariant systems and multi-agent systems. We first provide a general characterization of zero-dynamics attacks, which characterizes fully undetectable attacks targeting the non-minimum phase zeros of a system. This geometrical characterization makes it possible to develop a defense strategy seeking to place a minimal number of sensors to detect and counter the zero-dynamics attacks on the system's actuators. The detect and defense scheme amounts to computing a set containing potentially vulnerable actuator locations and nodes, and a defense union for feasible placement of sensors based on the geometrical properties of the cones under consideration.

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arXiv2021

On Uninformative Optimal Policies in Adaptive LQR with Unknown B-Matrix

This paper presents local asymptotic minimax regret lower bounds for adaptive Linear Quadratic Regulators (LQR). We consider affinely parametrized $B$-matrices and known $A$-matrices and aim to understand when logarithmic regret is impossible even in the presence of structural side information. After defining the intrinsic notion of an uninformative optimal policy in terms of a singularity condition for Fisher information we obtain local minimax regret lower bounds for such uninformative instances of LQR by appealing to van Trees' inequality (Bayesian Cramér-Rao) and a representation of regret in terms of a quadratic form (Bellman error). It is shown that if the parametrization induces an uninformative optimal policy, logarithmic regret is impossible and the rate is at least order square root in the time horizon. We explicitly characterize the notion of an uninformative optimal policy in terms of the nullspaces of system-theoretic quantities and the particular instance parametrization.

math.OC↗

arXiv2021

Power Injection Attacks in Smart Distribution Grids with Photovoltaics

In order to protect smart distribution grids from intrusions, it is important to understand possible risks and impacts of attacks. We study the worst-case attack strategy of a power injection attack against the physical layer of a smart distribution grid with a high penetration of photovoltaic resources. We derive both the worst attack signal and worst attack location: The worst attack signal is a step function which switches its sign at the final stage, and the worst attack location is the node with the largest impedance to the grid substation. Numerical examples on a European benchmark model verify the developed results. Finally, both theoretical and numerical results are used to discuss feasible defense strategies against power injection attacks.

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arXiv2021

Asymptotic Security by Model-based Incident Handlers for Markov Decision Processes

This study investigates general model-based incident handler's asymptotic behaviors in time against cyber attacks to control systems. The attacker's and the defender's dynamic decision making is modeled as an equilibrium of a dynamic signaling game. It is shown that the defender's belief on existence of an attacker converges over time for any attacker's strategy provided that the stochastic dynamics of the control system is known to the defender. This fact implies that the rational behavior of the attacker converges to a harmless action as long as the defender possesses an effective counteraction. The obtained result supports the powerful protection capability achieved by model-based defense mechanisms.

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