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Bias estimation in sensor networks
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Abstract
This paper investigates the problem of estimating biases affecting relative state measurements in a sensor network. Each sensor measures the relative states of its neighbors and this measurement is corrupted by a constant bias. We analyse under what conditions on the network topology and the maximum number of biased sensors the biases can be correctly estimated. We show that for non-bipartite graphs the biases can always be determined even when all the sensors are corrupted, while for bipartite graphs more than half of the sensors should be unbiased to ensure the correctness of the bias estimation. If the biases are heterogeneous, then the number of unbiased sensors can be reduced to two. Based on these conditions, we propose some algorithms to estimate the biases.
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In this paper we investigate sufficient conditions
for consensus of double integrators interconnected under constant
directed graphs, under the condition that there exists a
rooted spanning tree. We assume that only relative position as
well as absolute own velocity measurements are available that is,
each agent disposes of its own velocity only as well as its position
relatively to that of its neighbours. In addition, it is assumed
that the relative position measurements are unreliable, in the
sense that they are affected by a constant bias. Under these
conditions, we provide a consensus algorithm which ensures
that the systems stabilize near a common equilibrium point.
The analysis is based on Lyapunov direct method and a recent
novel approach of analysis of networked systems that takes into
account both the synchronization and the collective behavior.
This paper studies the problem of estimation from relative measurements in a graph, where a vector indexed over the nodes has to be reconstructed from relative pairwise measurements of differences between the values of nodes connected by an edge. In order to model the heterogeneity and uncertainty of the measurements, we assume that measurements are affected by additive noise distributed according to a Gaussian mixture. In this original setup, we formulate the problem of computing the Maximum-Likelihood (ML) estimates and we propose two novel Expectation-Maximization (EM) algorithms for its solution. The main difference between the two algorithm is that one of them is distributed and the other is not. The former algorithm is said to be distributed because it allows each node to compute the estimate of its own value by using only information that is directly available at the node itself or from its immediate neighbors. We prove convergence of both algorithms and present numerical simulations to evaluate and compare their performance.
Two asynchronous distributed algorithms are presented for solving a linear equation of the form with at least one solution. The equation is simultaneously and asynchronously solved by m agents assuming that each agent knows only a subset of the rows of the partitioned matrix [ ], the estimates of the equation’s solution generated by its neighbors, and nothing more. Neighbor relationships are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays. The event time sequences of different agents are not assumed to be synchronized. It is shown that for any matrix-vector pair ( ) for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents’ event times, the algorithms cause all agents’ estimates to converge exponentially fast to the same solution to . The first algorithm requires a specific initialization step at each agent, and the second algorithm works for arbitrary initializations. Explicit expressions for convergence rates are provided, and a relation between local initializations and limiting consensus solutions is established, which is used to solve the least 2-norm solution.
We focus on securely estimating the state of a nonlin-ear dynamical system from a set of corrupted measurements for two classes of nonlinear systems, and propose a technique which enables us to perform secure state estimation for those systems. We then illustrate how the proposed nonlinear secure state estimation technique can be used to perform estimation in the cyber layer of interconnected power systems under cyber-physical attacks and communication failures. In particular, we focus on an interconnected power system comprising several synchronous generators, transmission lines, loads, and energy storage units, and propose a secure estimator that allows us to securely estimate the dynamic states of the power network. Finally, we numerically demonstrate the effectiveness of the proposed secure estimation algorithm, and show that the algorithm enables the cyber layer to accurately reconstruct the attack signals. Index Terms—Cyber-physical systems, secure state estimation, power systems, dynamic state estimation.
We consider the problem of distributed state estimation of a linear time-invariant (LTI) system by a network of sensors. We develop a distributed observer that guarantees asymptotic reconstruction of the state for the most general class of LTI systems, sensor network topologies and sensor measurement structures. Our analysis builds upon the following key observation - a given node can reconstruct a portion of the state solely by using its own measurements and constructing appropriate Luenberger observers; hence it only needs to exchange information with neighbors (via consensus dynamics) for estimating the portion of the state that is not locally detectable. This intuitive approach leads to a new class of distributed observers with several appealing features. Furthermore, by imposing additional constraints on the system dynamics and network topology, we show that it is possible to construct a simpler version of the proposed distributed observer that achieves the same objective while admitting a fully distributed implementation.
Cyber-physical systems (CPSs) are found in many applications such as power
networks, manufacturing processes, and air and ground transportation systems.
Maintaining security of these systems under cyber attacks is an important and
challenging task, since these attacks can be erratic and thus difficult to
model. Secure estimation problems study how to estimate the true system states
when measurements are corrupted and/or control inputs are compromised by
attackers. The authors in [1] proposed a secure estimation method when the set
of attacked nodes (sensors, controllers) is fixed. In this paper, we extend
these results to scenarios in which the set of attacked nodes can change over
time. We formulate this secure estimation problem into the classical error
correction problem [2] and we show that accurate decoding can be guaranteed
under a certain condition. Furthermore, we propose a combined secure estimation
method with our proposed secure estimator above and the Kalman Filter (KF) for
improved practical performance. Finally, we demonstrate the performance of our
method through simulations of two scenarios where an unmanned aerial vehicle is
under adversarial attack.
Despite the great success of using gradient-based controllers to stabilize rigid formations of autonomous agents in the past years, surprising yet intriguing undesirable collective motions have been reported recently, when inconsistent measurements are used in the agents' local controllers. To make the existing gradient control robust against such measurement inconsistency, we exploit local estimators following the well-known internal model principle for robust output regulation control. The new estimator-based gradient control is still distributed in nature, and can be constructed systematically even when the number of agents in a rigid formation grows. We prove rigorously that the proposed control is able to guarantee exponential convergence, and then demonstrate through robotic experiments and computer simulations that the reported inconsistency-induced orbits of collective movements are effectively eliminated.
We propose a distributed algorithm for solving the optimization problem Basis Pursuit (BP). BP finds the least ℓ1-norm solution of the underdetermined linear system Ax = b and is used, for example, in compressed sensing for reconstruction. Our algorithm solves BP on a distributed platform such as a sensor network, and is designed to minimize the communication between nodes. The algorithm only requires the network to be connected, has no notion of a central processing node, and no node has access to the entire matrix A at any time. We consider two scenarios in which either the columns or the rows of A are distributed among the compute nodes. Our algorithm, named D-ADMM, is a decentralized implementation of the alternating direction method of multi- pliers. We show through numerical simulation that our algorithm requires considerably less communications between the nodes than the state-of-the-art algorithms.
This paper describes two algorithms for state reconstruction from sensor
measurements that are corrupted with sparse, but otherwise arbitrary, "noise".
These results are motivated by the need to secure cyber-physical systems
against a malicious adversary that can arbitrarily corrupt sensor measurements.
The first algorithm reconstructs the state from a batch of sensor measurements
while the second algorithm is able to incorporate new measurements as they
become available, in the spirit of a Luenberger observer. A distinguishing
point of these algorithms is the use of event-triggered techniques to
circumvent some limitations that arise when performing state reconstruction
with sparse signals.
This paper proposes a fully decentralized adaptive re-weighted state
estimation (DARSE) scheme for power systems via network gossiping. The enabling
technique is the proposed Gossip-based Gauss-Newton (GGN) algorithm, which
allows to harness the computation capability of each area (i.e. a database
server that accrues data from local sensors) to collaboratively solve for an
accurate global state. The DARSE scheme mitigates the influence of bad data by
updating their error variances online and re-weighting their contributions
adaptively for state estimation. Thus, the global state can be estimated and
tracked robustly using near-neighbor communications in each area. Compared to
other distributed state estimation techniques, our communication model is
flexible with respect to reconfigurations and resilient to random failures as
long as the communication network is connected. Furthermore, we prove that the
Jacobian of the power flow equations satisfies the Lipschitz condition that is
essential for the GGN algorithm to converge to the desired solution.
Simulations of the IEEE-118 system show that the DARSE scheme can estimate and
track online the global power system state accurately, and degrades gracefully
when there are random failures and bad data.
In this paper, we address the problem of simultaneous classification and
estimation of hidden parameters in a sensor network with communications
constraints. In particular, we consider a network of noisy sensors which
measure a common scalar unknown parameter. We assume that a fraction of the
nodes represent faulty sensors, whose measurements are poorly reliable. The
goal for each node is to simultaneously identify its class (faulty or
non-faulty) and estimate the common parameter.
We propose a novel cooperative iterative algorithm which copes with the
communication constraints imposed by the network and shows remarkable
performance. Our main result is a rigorous proof of the convergence of the
algorithm and a characterization of the limit behavior. We also show that, in
the limit when the number of sensors goes to infinity, the common unknown
parameter is estimated with arbitrary small error, while the classification
error converges to that of the optimal centralized maximum likelihood
estimator. We also show numerical results that validate the theoretical
analysis and support their possible generalization. We compare our strategy
with the Expectation-Maximization algorithm and we discuss trade-offs in terms
of robustness, speed of convergence and implementation simplicity.
The ability to determine absolute distance to an object is one of the most basic measurements of remote sensing. High-precision ranging has important applications in both large-scale manufacturing and in future tight formation-flying satellite missions, where rapid and precise measurements of absolute distance are critical for maintaining the relative pointing and position of the individual satellites. Using two coherent broadband fibre-laser frequency comb sources, we demonstrate a coherent laser ranging system that combines the advantages of time-of-flight and interferometric approaches to provide absolute distance measurements, simultaneously from multiple reflectors, and at low power. The pulse time-of-flight yields a precision of 3νm with an ambiguity range of 1.5m in 200νs. Through the optical carrier phase, the precision is improved to better than 5nm at 60ms, and through the radio-frequency phase the ambiguity range is extended to 30km, potentially providing 2 parts in 10 13 ranging at long distances.
In this paper we study the problem of estimating the channel parameters for a generic wireless sensor network (WSN) in a completely distributed manner, using consensus algorithms. Specifically, we first propose a distributed strategy to minimize the effects of unknown constant offsets in the reading of the radio strength signal indicator due to uncalibrated sensors. Then we show how the computation of the optimal wireless channels parameters, which are the solution of a global least-square optimization problem, can be obtained with a consensus-based algorithm. The proposed algorithms are general algorithms for sensor calibration and distributed least-square parameter identification, and do not require any knowledge either on the global topology of the network nor the total number of nodes. Finally, we apply these algorithms to experimental data collected from an indoor WSN.
Large-scale sensor networks give rise to estimation problems that have a rich graphical structure. We studied one of these problems in terms of how such an estimate can be efficiently computed in a distributed manner as well as how the quality of an optimal estimate scales with the size of the network. Two distributed algorithms are presented to compute the optimal estimates that are scalable and robust to communication failures. In designing these algorithms, we found the literature on parallel computation to be a rich source of inspiration.
In this paper, we propose a distributed state-and-fault estimation scheme for multi-agent systems. The estimator is based on an
-norm optimization problem, which is inspired by sparse signal recovery in the field of compressive sampling. Two theoretical results are given to analyze the correctness of our approach. First, we provide a necessary and sufficient condition such that the state and fault signals are correctly estimated. The result presents a fundamental limitation of the algorithm, which shows how many faulty nodes are allowed to ensure a correct estimation. Second, we analyze how the estimation error grows over time by showing that the upper bound of the estimation error depends on the previous state estimate and the number of faulty nodes. An illustrative example is given to validate the effectiveness of the proposed approach.
The problem of identifying sparse solutions for the link structure and dynamics of an unknown linear, time-invariant network is posed as finding sparse solutions to . If the matrix satisfies a rank condition, this problem has a unique, sparse solution. Here each row of comprises one experiment consisting of input/output measurements and cannot be freely chosen. We show that if experiments are poorly designed, the rank condition may never be satisfied, resulting in multiple solutions. We discuss strategies for designing experiments such that has the desired properties and the problem is therefore well posed. This formulation allows prior knowledge to be taken into account in the form of known nonzero entries of , requiring fewer experiments to be performed. Simulated examples are given to illustrate the approach, which provides a useful strategy commensurate with the type of experiments and measurements available to biologists. We also confirm suggested limitations on the use of convex relaxations for the efficient solution of this problem.
This paper studies the relative position based node-localization problem for a sensor network without all nodes sharing a common reference frame in the presence of both measurement and communication noises. To solve the problem, a robust distributed orientation estimate algorithm and a robust distributed node-localization algorithm are designed, where unbiased estimators are constructed based on the historical measurement information to inhibit the measurement noise and the stochastic approximation method is adopted to inhibit the communication noise. Under the zero-mean and independence assumption on the measurement/communication noise, we show that all sensor nodes can asymptotically determine their own orientation angles and positions almost surely under the designed algorithms, if and only if the network contains at least one anchor node, and its communication and distance-sensing topology is 1-rooted at the anchor node set and the corresponding bearing sensing topology is connected. Moreover, the convergence rate is quantified if only the measurement noise or communication noise is involved. Simulation experiments are conducted to validate the effectiveness of the proposed algorithms.
Fifth-generation (5G) networks providing much higher bandwidth and faster data rates will allow connecting vast number of stationary and mobile devices, sensors, agents, users, machines, and vehicles, supporting Internet-of-Things (IoT), real-time dynamic networks of mobile things. Positioning and location awareness will become increasingly important, enabling deployment of new services and contributing to significantly improving the overall performance of the 5G system. Many of the currently talked about solutions to positioning in 5G are centralized, mostly requiring direct communication to the access nodes (or anchors, i.e., nodes with known locations), which in turn requires a high density of anchors. But such centralized positioning solutions may become unwieldy as the number of users and devices continues to grow without limit in sight. As an alternative to the centralized solutions, this paper discusses distributed localization in a 5G-enabled IoT environment where many low power devices, users, or agents are to locate themselves without a direct access to anchors. Even though positioning is essentially a nonlinear problem (solving circle equations by trilateration or triangulation), we discuss a cooperative linear distributed iterative solution with only local measurements, local communication, and local computation needed at each agent. Linearity is obtained by reparametrization of the agent location through barycentric coordinate representations based on local neighborhood geometry that may be computed in terms of certain Cayley-Menger determinants involving relative local inter-agent distance measurements. After a brief introduction to the localization problem, and other available distributed solutions primarily based on directly addressing the nonlinear formulation, we present the distributed linear solution for stationary agent networks and study its convergence, its robustness to noise, and extensions to mobile scenarios, in which agents, users, and (possibly) anchors are dynamic.
In this paper we consider a novel partitioned framework for distributed optimization in peer-to-peer networks. In several important applications the agents of a network have to solve an optimization problem with two key features: (i) the dimension of the decision variable depends on the network size, and (ii) cost function and constraints have a sparsity structure related to the communication graph. For this class of problems a straightforward application of existing consensus methods would show two inefficiencies: poor scalability and redundancy of shared information. We propose an asynchronous distributed algorithm, based on dual decomposition and coordinate methods, to solve partitioned optimization problems. We show that, by exploiting the problem structure, the solution can be partitioned among the nodes, so that each node just stores a local copy of a portion of the decision variable (rather than a copy of the entire decision vector) and solves a small-scale local problem.
In this paper we address the problem of fault resilient estimation for large-scale systems, where the measurements are possibly corrupted due to faults of low-cost sensors. As a toy application, we consider the problem of localization in Sensor Networks (SN). We propose a distributed solution based on a recently developed generalized descent algorithm. To cope with real-world applications, the algorithm we propose is suitable for an asynchronous implementation and is numerically robust to non ideal communications, i.e., packet-losses. Under mild assumptions, theoretical convergence of the algorithm is shown. The algorithm is compared with a recently developed ADMM-based algorithm for robust state estimation.
In this paper, we propose a novel distributed formation control strategy, which is based on the measurements of relative position of neighbors, with global orientation estimation. Since agents do not share a common reference frame, orientations of the local reference frames are not aligned with each other. The proposed strategy includes a combination of global orientation estimation and formation control law. Under the proposed strategy, the orientation of each agent is estimated in the global sense, if interaction graph has a spanning tree. With the estimated orientations of local frames, formation control strategy ensures that the formation globally exponentially converges to the desired formation.
This paper presents a secure and robust state estimation scheme for continuous-time linear dynamical systems. The method is secure in that it correctly estimates the states under sensor attacks by exploiting sensing redundancy, and it is robust in that it guarantees a bounded estimation error despite measurement noises and process disturbances. In this method, an individual Luenberger observer (of possibly smaller size) is designed from each sensor. Then, the state estimates from each of the observers are combined through a scheme motivated by error correction techniques, which results in estimation resiliency against sensor attacks under a mild condition on the system observability. Moreover, in the state estimates combining stage, our method reduces the search space of a minimization problem to a finite set, which substantially reduces the required computational effort.
This paper addresses the problem of bearing-based network localization, which aims to localize all the nodes in a static network given the locations of a subset of nodes termed anchors and inter-node bearings measured in a common reference frame. The contributions of the paper are twofold. Firstly, we propose necessary and sufficient conditions for network localizability with both algebraic and rigidity theoretic interpretations. Secondly, we propose and analyze a linear distributed protocol for bearing-based network localization. One novelty of our work is that the localizability analysis and localization protocol are applicable to networks in arbitrary dimensional spaces.
This article addresses the formation control problem with mismatched compasses. Depending on the sensing and communication technology, compass mismatches may arise due to biases in measurement, drift in inertial sensing despite initial alignment, and even spatial variations in the earth’s magnetic field. To illustrate the key concepts underlying what happens, we first consider the two agent case and show that the agents converge to a fixed, but distorted formation exponentially fast. In contrast to the matched compass case, the formation is not asymptotically stationary. The distance error and the angular error between the actual final formation and the desired formation are explicitly given, as is the steady state velocity of the formation. The case of time-varying mismatched compasses is also studied. Based on the results, we then propose estimators to obtain the mismatched angle, which allow a compensation algorithm to be proposed such that the desired formation shape is achieved. Finally, the extensions to the agent case are also considered and similar phenomena are encountered. Simulations are provided to validate the theoretical results.
This technical note analyzes the effect of stealthy integrity attacks on Cyber-Physical Systems, which is modeled as a Stochastic Linear Time-Invariant (LTI) system equipped with a linear filter, a linear feedback controller and a χ2 failure detector. An attacker wishes to induce perturbation in the control loop by compromising a subset of the sensors and injecting an exogenous control input, without incurring detection from an anomaly detector. We show how the problem can be modeled, from the attacker's standpoint, as a constrained control problem and that the characterization of the maximum perturbation can be posed as reachable set computation, which we solve using ellipsoidal calculus.
We study distributed network flows as solvers in continuous time for the
linear algebraic equation . Each node i has
access to a row of the matrix and the
corresponding entry in the vector . The first "consensus +
projection" flow under investigation consists of two terms, one from standard
consensus dynamics and the other contributing to projection onto each affine
subspace specified by the and . The second "projection
consensus" flow on the other hand simply replaces the relative state feedback
in consensus dynamics with projected relative state feedback. Without
dwell-time assumption on switching graphs as well as without positively lower
bounded assumption on arc weights, we prove that all node states converge to a
common solution of the linear algebraic equation, if there is any. The
convergence is global for the "consensus + projection" flow while local for the
"projection consensus" flow in the sense that the initial values must lie on
the affine subspaces. If the linear equation has no exact solutions, we show
that the node states can converge to a ball around the least squares solution
whose radius can be made arbitrarily small through selecting a sufficiently
large gain for the "consensus + projection" flow under fixed bidirectional
graphs. Semi-global convergence to approximate least squares solutions is
demonstrated for general switching directed graphs under suitable conditions.
It is also shown that the "projection consensus" flow drives the average of the
node states to the least squares solution with complete graph. Numerical
examples are provided as illustrations of the established results.
The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries-stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l(1) norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.
The emerging ad hoc clouds form a new cloud computing paradigm by leveraging
untapped local computation and storage resources. An important application over
ad hoc clouds is outsourcing computationally intensive problems to nearby cloud
agents to solve in a distributed manner. A risk with ad hoc clouds is however
the potential cyber attacks, with the security and privacy in distributed
outsourcing being a significant challenging issue. In this paper, we consider
distributed secure outsourcing of linear algebraic equations (LAE), one of the
most frequently used mathematical tool, in ad hoc clouds. The outsourcing
client assigns each agent a subproblem; all involved agents then apply a
consensus based algorithm to obtain the correct solution in a distributed and
iterative manner. We identify a number of security risks in this process, and
propose a secure outsourcing scheme which can not only preserve privacy to
shield the original LAE parameters and the final solution from the computing
agents, but also detect misbehavior based on mutual verifications in a
real-time manner. We rigorously prove that the proposed scheme converges to the
correct solution of the LAE exponentially fast, has low computation complexity
at each agent, and is robust against the identified security attacks. Extensive
numerical results are presented to demonstrate the effectiveness of the
proposed method.
This paper presents a robust, distributed algorithm to solve general linear
programs. The algorithm design builds on the characterization of the solutions
of the linear program as saddle points of a modified Lagrangian function. We
show that the resulting continuous-time saddle-point algorithm is provably
correct but, in general, not distributed because of a global parameter
associated with the nonsmooth exact penalty function employed to encode the
inequality constraints of the linear program. This motivates the design of a
discontinuous saddle-point dynamics that, while enjoying the same convergence
guarantees, is fully distributed and scalable with the dimension of the
solution vector. We also characterize the robustness against disturbances and
link failures of the proposed dynamics. Specifically, we show that it is
integral-input-to-state stable but not input-to-state stable. The latter fact
is a consequence of a more general result, that we also establish, which states
that no algorithmic solution for linear programming is input-to-state stable
when uncertainty in the problem data affects the dynamics as a disturbance. Our
results allow us to establish the resilience of the proposed distributed
dynamics to disturbances of finite variation and recurrently disconnected
communication among the agents. Simulations in an optimal control application
illustrate the results.
In this paper, we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors by means of only local communication and bounded complexity, independent of network size and topology. We propose a consensus-based algorithm with the use of local memory variables which allows asynchronous implementation, has guaranteed exponential convergence to the optimal solution under simple deterministic and randomized communication protocols, and requires minimal packet transmission. In the randomized scenario, we then study the rate of convergence in expectation of the estimation error and we argue that it can be used to obtain upper and lower bound for the rate of converge in mean square. In particular, we show that for regular graphs, such as Cayley, Ramanujan, and complete graphs, the convergence rate in expectation has the same asymptotic degradation of memoryless asynchronous consensus algorithms in terms of network size. In addition, we show that the asynchronous implementation is also robust to delays and communication failures. We finally complement the analytical results with some numerical simulations, comparing the proposed strategy with other algorithms which have been recently proposed in the literature.
We propose a formation control strategy based on inter-agent displacements for single-integrator modeled agents in the plane. Since the orientations of the local reference frames of the agents are not aligned with each other due to the absence of a common sense of orientation, the proposed strategy consists of an orientation alignment law and a formation control law. Under the proposed strategy, if the interaction graph is uniformly connected and all the initial orientation angles belong to an interval with length less than pi , the orientations are exponentially aligned and the formation exponentially converges to the desired formation. We also show that the proposed strategy can be utilized for network localization as a dual problem.
Graph theory has been used to characterize the solvability of the sensor network localization problem with ideal (i.e., precisely known) bearing-only measurements between certain pairs of sensors and a limited amount of information about the position of certain nodes, i.e., anchors. In practice, however, bearing measurements will never be exact, and the equations whose solutions deliver sensor positions in the noiseless case may no longer have a solution. This technical brief argues that if the same conditions for localizability that exist in the noiseless case are satisfied and the bearing measurement errors are small enough (as will be formalized later in the technical brief), then the network will be approximately localizable, i.e., sensor position estimates can be found which are near the correct values. In particular, a bound on the position errors is found in terms of a bound on the bearing errors. Later, this bound is used to propose a method to select anchors to minimize the effect of noisy bearing measurements on the localization solution.
Deregulation of energy markets, penetration of renewables, advanced metering capabilities, and the urge for situational awareness, all call for system-wide power system state estimation (PSSE). Implementing a centralized estimator though is practically infeasible due to the complexity scale of an interconnection, the communication bottleneck in real-time monitoring, regional disclosure policies, and reliability issues. In this context, distributed PSSE methods are treated here under a unified and systematic framework. A novel algorithm is developed based on the alternating direction method of multipliers. It leverages existing PSSE solvers, respects privacy policies, exhibits low communication load, and its convergence to the centralized estimates is guaranteed even in the absence of local observability. Beyond the conventional least-squares based PSSE, the decentralized framework accommodates a robust state estimator. By exploiting interesting links to the compressive sampling advances, the latter jointly estimates the state and identifies corrupted measurements. The novel algorithms are numerically evaluated using the IEEE 14-, 118-bus, and a 4200-bus benchmarks. Simulations demonstrate that the attainable accuracy can be reached within a few inter-area exchanges, while largest residual tests are outperformed.
In this paper we propose a novel distributed algorithm to solve degenerate linear programs on asynchronous peer-to-peer networks with distributed information structures. We propose a distributed version of the well-known simplex algorithm for general degenerate linear programs. A network of agents, running our algorithm, will agree on a common optimal solution, even if the optimal solution is not unique, or will determine infeasibility or unboundedness of the problem. We establish how the multi-agent assignment problem can be efficiently solved by means of our distributed simplex algorithm. We provide simulations supporting the conjecture that the completion time scales linearly with the diameter of the communication graph.
The vast majority of today's critical infrastructure is supported by numerous
feedback control loops and an attack on these control loops can have disastrous
consequences. This is a major concern since modern control systems are becoming
large and decentralized and thus more vulnerable to attacks. This paper is
concerned with the estimation and control of linear systems when some of the
sensors or actuators are corrupted by an attacker. In the first part we look at
the estimation problem where we characterize the resilience of a system to
attacks and study the possibility of increasing its resilience by a change of
parameters. We then propose an efficient algorithm to estimate the state
despite the attacks and we characterize its performance. Our approach is
inspired from the areas of error-correction over the reals and compressed
sensing. In the second part we consider the problem of designing
output-feedback controllers that stabilize the system despite attacks. We show
that a principle of separation between estimation and control holds and that
the design of resilient output feedback controllers can be reduced to the
design of resilient state estimators.
We analyze the spatial smoothing algorithm of Solis, Borkar and Kumar (2005) for clock synchronization over multi-hop wireless networks. In particular, for a model of a random wireless network we show that with high probability the error variance is O(1) as the number of nodes in the network increases. This provides support for the feasibility of time-based computing n large wireless networks. We also provide bounds on the settling time of a distributed algorithm
We consider the problem of estimating a state x from noisy and corrupted linear measurements y = Ax + z + e, where z is a dense vector of small-magnitude noise and e is a relatively sparse vector whose entries can be arbitrarily large. We study the behavior of the lscr1 estimator xcirc = arg minx ||y - Ax||1, and analyze its breakdown point with respect to the number of corrupted measurements ||e||0. We show that the breakdown point is independent of the noise. We introduce a novel algorithm for computing the breakdown point for any given A, and provide a simple bound on the estimation error when the number of corrupted measurements is less than the breakdown point. As a motivational example we apply our algorithm to design a robust state estimator for an autonomous vehicle, and show how it can significantly improve performance over the Kalman filter.
We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements.
This paper proposes a simple, distributed algorithm that achieves global stabilization of formations for relative sensing networks in arbitrary dimensions with fixed topology. Assuming the network runs an initialization procedure to equally orient all agent reference frames, convergence to the desired formation shape is guaranteed even in partially asynchronous settings. We characterize the algorithm robustness against several sources of errors: link failures, measurement errors, and frame initialization errors. The technical approach combines algebraic graph theory, multidimensional scaling, and distributed linear iterations.
This paper considers dynamic laws that seek a saddle point of a function of two vector variables, by moving each in the direction of the corresponding partial gradient. This method has old roots in the classical work of Arrow, Hurwicz and Uzawa on convex optimization, and has seen renewed interest with its recent application to resource allocation in communication networks. This paper brings other tools to bear on this problem, in particular Krasovskii’s method to find Lyapunov functions, and recently obtained extensions of the LaSalle invariance principle for hybrid systems. These methods are used to obtain stability proofs of these primal–dual laws in different scenarios, and applications to cross-layer network optimization are exhibited.
Conventional approaches to sampling signals or images follow Shannon's theorem: the sampling rate must be at least twice the maximum frequency present in the signal (Nyquist rate). In the field of data conversion, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation - the signal is uniformly sampled at or above the Nyquist rate. This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisition. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than traditional methods use.
This paper considers a natural error correcting problem with real valued input/output. We wish to recover an input vector f∈Rn from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1-minimization problem (||x||ℓ1:=Σi|xi|) min(g∈Rn) ||y - Ag||ℓ1 provided that the support of the vector of errors is not too large, ||e||ℓ0:=|{i:ei ≠ 0}|≤ρ·m for some ρ>0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of ℓ1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.
Distributed asynchronous algorithms for solving positive definite linear equations over networkspart i: Agent networks
- J Lu
- C Y Tang
J. Lu and C. Y. Tang, "Distributed asynchronous algorithms for solving
positive definite linear equations over networkspart i: Agent networks,"
IFAC Proceedings Volumes, vol. 42, no. 20, pp. 252-257, 2009.
A mathematical introduction to compressive sensing
- S Foucart
- H Rauhut
S. Foucart and H. Rauhut, A mathematical introduction to compressive
sensing. Birkhäuser Basel, 2013, vol. 1, no. 3.
A robust blockjacobi algorithm for quadratic programming under lossy communications
- M Todescato
- G Cavraro
- R Carli
- L Schenato
M. Todescato, G. Cavraro, R. Carli, and L. Schenato, "A robust blockjacobi algorithm for quadratic programming under lossy communications," IFAC-PapersOnLine, vol. 48, no. 22, pp. 126-131, 2015.