Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> This paper provides a theoretical framework for analysis of consensus algorithms for multi-agent networked systems with an emphasis on the role of directed information flow, robustness to changes in network topology due to link/node failures, time-delays, and performance guarantees. An overview of basic concepts of information consensus in networks and methods of convergence and performance analysis for the algorithms are provided. Our analysis frame- work is based on tools from matrix theory, algebraic graph theory, and control theory. We discuss the connections between consensus problems in networked dynamic systems and diverse applications including synchronization of coupled oscillators, flocking, formation control, fast consensus in small- world networks, Markov processes and gossip-based algo- rithms, load balancing in networks, rendezvous in space, distributed sensor fusion in sensor networks, and belief propagation. We establish direct connections between spectral and structural properties of complex networks and the speed of information diffusion of consensus algorithms. A brief introduction is provided on networked systems with nonlocal information flow that are considerably faster than distributed systems with lattice-type nearest neighbor interactions. Simu- lation results are presented that demonstrate the role of small- world effects on the speed of consensus algorithms and cooperative control of multivehicle formations. <s> BIB001 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> A vehicle suspension assembly for tandem drive axles has a suspension system for each axle. Each suspension system incorporates two lower torque beams on each side of the vehicle and a single central upper torque rod. A central hanger is mounted between the two axles. The lower torque beams are pivotally connected between the hanger and an axle seat. A straddle bracket connects the axle to the axle seat in a hinged fashion. The upper central torque rod is connected between the upper side of the axle housing and the chassis. The lower torque beams act as a parallelogram to stabilize the axle seat. They also act together with a portion of the axle seat as a composite lower torque rod cooperating with the upper torque rod to stabilize the axle. <s> BIB002 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> We propose CaliBree, a self-calibration system for mobile wireless sensor networks. Sensors calibration is a fundamental problem in a sensor network. If sensor devices are not properly calibrated, their sensor readings are likely of little use to the application. Distributed calibration is challenging in a mobile sensor network, where sensor devices are carried by people or vehicles, mainly for three reasons: i)the sensing contact time, i.e., the amount of time nodes are within the sensing range of each other, can be very limited, requiring a quick and efficient calibration technique; ii)for scalability and ease of use, the calibration algorithm should not require manual intervention; iii)the computational burden must be low since some sensor platforms have limited capabilities. In this paper we propose CaliBree, a distributed, scalable, and lightweight calibration protocol that applies a discrete average consensus algorithm technique to calibrate sensor nodes. CaliBree is shown to be effective through experimental evaluation using embedded wireless sensor devices, achieving high calibration accuracy. <s> BIB003 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> 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. Copyright © 2009 John Wiley & Sons, Ltd. <s> BIB004 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> In this paper a novel consensus-based distributed algorithm for blind macro-calibration in sensor networks is proposed. It is proved, on the basis of an originally developed methodology for treating higher order consensus schemes, that the algorithm achieves asymptotic agreement for sensor gains and offsets in the mean square sense and with probability one. In the case of a given reference, all sensors are asymptotically calibrated. Simulation results illustrate properties of the algorithm. <s> BIB005 </s> Self-Calibration Methods for Uncontrolled Environments in Sensor Networks: A Reference Survey <s> Consensus Algorithms <s> A novel distributed algorithm for blind macro-calibration of large sensor networks is introduced. The algorithm is in the form of a system of gradient-type recursions for estimating parameters of local sensor calibration functions. The method does not require any fusion center. The convergence analysis is based on diagonal dominance of the dynamical systems with block matrices. It is proved that the asymptotic consensus is achieved for all the equivalent sensor gains and offsets (in the mean square sense and with probability one) in lossy sensor networks with possible communication outages and additive communication noise. An illustrative simulation example is provided. <s> BIB006
The consensus problem goal in a distributed system with multiple agents is the agreement among the agents for a single data value. In the case of calibration, a network of uncalibrated nodes interacts with calibrated nodes to solve a consensus problem. The average consensus algorithm measures the sensor samples' disagreement between the uncalibrated node and a set of calibrated neighbors. The algorithm converges to a consensus value among the set of nodes and leads to the discovery of the actual disagreement between the uncalibrated node's sensor and calibrated nodes' sensors. In a consensus algorithm, BIB001 , the state of agent n in graph G=(V, E) is denoted as x n . A consensus algorithm to reach an agreement regarding the state of n agents with dynamicsẋ i =u i , whereẋ denotes derivative, can be written as an nth-order linear system: with x i (0)=z i ∈R and b i (0)=0. It can be shown BIB001 that the iterative discrete consensus algorithm can be expressed as CaliBree BIB003 is a model-based calibration network in which ground-truth calibrated nodes inform non-calibrated nodes, using a beacon protocol, that they will participate in the calibration process. These nodes, then, use a discrete consensus distributed algorithm to calibrate the uncalibrated node by computing the degree of disagreement between the reference nodes and the uncalibrated node. The nodes are separated in two sets: nodes that are calibrated and nodes that are not calibrated. Let N i be the neighboring set of calibrated nodes of an uncalibrated node i. The state, denoted as s i (k), is the data sample taken at time k, andd i (k) is the average disagreement measured by node i up to round k. The disagreement is defined as the difference between the state and the consensus value, that is, , is the difference between the uncalibrated node i and a calibrated node. The formulation of the disagreement consensus problem for calibration of data is then BIB003  CaliBree BIB003 considered that neighboring calibrated nodes periodically send beacons with their ground-truth or calibrated sensed data. Uncalibrated nodes run the consensus algorithm to achieve a consensus value, showing that the algorithm converges to a minimum disagreementd min i . The CaliBree protocol performance is tested using an uncalibrated mobile light sensor node that comes in the range of calibrated sensor nodes. In a similar work, Bolognani et al. BIB004 proposed a distributed algorithm to micro-calibrate the radio strength signal indicator (RSSI) that is applied to localization and tracking for WSNs. The algorithm uses mean squares to estimate wireless channel parameters, whereas a consensus algorithms is introduced to estimate the sensor offset that affects the measured received strength of the receiving node due to fabrication mismatches in the radio chip and that can produce ±6dB in RSSI measurements. Stankovic et al. BIB006 proposed a distributed non-blind or blind macro-calibration mechanism to calibrate the coefficients of a network of N sensors measuring a common discrete signal {x(t)} that is considered a random process. The algorithm can be considered a non-trivial extension of consensus algorithms. Each sensor n produces an output signal y n (t) = β 0n + β 1n x(t). The goal is to achieve a calibration function z n (t) = a n y n + b n = a n β 1n x(t) + a n β 0n + b n = g n x(t) + f n so that when there is no ground-truth reference signal (distributed blind macro-calibration), the sensor produces asymptotically equal outputs z n (t). The mechanism may also be adapted to the case in which a sensor gives ground-truth reference data (distributed non-blind macro-calibration), producing a gain g n =1 and offset f n =0 for all n=1, ..., N. If θ n =[a n , b n ]. The distributed blind calibration algorithm minimizes the instantaneous difference between the signal measured by a sensor and its neighborhood: where γ i j are nonnegative scalar weights reflecting the relative importance of the neighboring nodes. If we apply a stochastic gradient descent algorithm, the parameter θ n is expressed aŝ where δ(t) is the step length at each step t of the algorithm. The algorithm has to begin with an initial point θ(0)= BIB002 0] . At each step, node n has to receive the updates of its neighboring nodes. Stankovic et al. BIB005 gave conditions for the convergence of the algorithm and showed how to modify the algorithm in case a sensor acts as a ground-truth sensor, giving reference data to its neighbors.