Optimal Sensor Placement Problem Research Articles

Graph Laplace Regularization-based pressure sensor placement strategy for leak localization in the water distribution networks under joint hydraulic and topological feature spaces

Urban water distribution networks (WDNs) have wide range and intricate topology, which include leakage, pipe burst and other abnormal states during production and operation. With the continuous development of the Internet of Things (IoT) technology in recent years, the means of monitoring the WDNs by using wireless sensor network technology has gradually received attention and extensive research. Most of the existing researches select the deployment location of sensors according to the hydraulic state of the WDNs, but the connectivity and topology between the nodes of the WDNs are not fully considered and analyzed. In this study, a new method that can integrate the topological features and hydraulic model information of the WDN is proposed to solve the problem of optimal sensor placement. First, the method preprocesses the covariance matrix of the pressure sensitivity matrix of the water distribution network by a diffusion kernel-based data prefiltering method and obtains the new network topology weights and its Laplacian matrix under the constraints of the network topology through a data-based graphical Laplacian learning method. Then, the sensor placement problem is transformed into a matrix minimum eigenvalue constraint problem by the Graph Laplace Regularization (GLR)-based method, and finally the selection of sensor nodes is accomplished by the method based on Gershgorin Disc Alignment (GDA). The proposed strategy is tested on a passive Hanoi network, an active Net 3 network, and a larger network, PA2, and is compared with some existing methods. The results show that the proposed solution achieves good performance in three different leak localization methods.

State Observer-Based Conditioned Reverse-Path Method for Nonlinear System Identification

In light of the complex behavior of vibrating structures, their reliable modeling plays a crucial role in the analysis and system design for vibration control. In this paper, the reverse-path (RP) method is revisited, further developed, and applied to modeling a nonlinear system, particularly with respect to the identification of the frequency response function for a nominal underlying linear system and the determination of the structural nonlinearities. The present approach aims to overcome the requirement for measuring all nonlinear system states all the time during operation. Especially in large-scale systems, this might be a tedious task and often practically infeasible since it would require having individual sensors assigned for each state involved in the design process. In addition, the proper placement and simultaneous operation of a large number of transducers would represent further difficulty. To overcome those issues, we have proposed state estimation in light of the observability criteria, which significantly reduces the number of required sensor elements. To this end, relying on the optimal sensor placement problem, the state estimation process reduces to the solution of Kalman filtering. On this ground, the problem of nonlinear system identification for large-scale systems can be addressed using the observer-based conditioned RP method (OBCRP) proposed in this paper. In contrast to the classical RP method, the current one can potentially handle local and distributed nonlinearities. Moreover, in addition to the state estimation and in comparison to the orthogonal RP method, a new frequency-dependent weighting is introduced in this paper, which results in superior nonlinear system identification performances. Implementation of the method is demonstrated on a multi-degree-of-freedom discretized lumped mass system, representing a substitute model of a physical counterpart used for the identification of the model parameters.

A Novel Optimal Sensor Placement Method for Optimizing the Diagnosability of Liquid Rocket Engine

There are hundreds of various sensors used for online Prognosis and Health Management (PHM) of LREs. Inspired by the fact that a limited number of key sensors are selected for inflight control purposes in LRE, it is practical to optimal placement of redundant sensors for improving the diagnosability and economics of PHM systems. To strike a balance between sensor cost, real-time performance and diagnosability of the fault diagnosis algorithm in LRE, this paper proposes a novel Optimal Sensor Placement (OSP) method. Firstly, a Kernel Extreme Learning Machine-based (KELM) two-stage diagnosis algorithm is developed based on a system-level failure simulation model of LRE. Secondly, hierarchical diagnosability metrics are constructed to formulate the OSP problem in this paper. Thirdly, a Hierarchy Ranking Evolutionary Algorithm-based (HREA) two-stage OSP method is developed, achieving further optimization of Pareto solutions by the improved hypervolume indicator. Finally, the proposed method is validated using failure simulation datasets and hot-fire test-run experiment datasets. Additionally, four classical binary multi-objective optimization algorithms are introduced for comparison. The testing results demonstrate that the HREA-based OSP method outperforms other classical methods in effectively balancing the sensor cost, real-time performance and diagnosability of the diagnosis algorithm. The proposed method in this paper implements system-level OSP for LRE fault diagnosis and exhibits the potential for application in the development of reusable LREs.

Leveraging Swarm Intelligence for Optimal Thermal Camera and Sensor Placement in Industrial Environments

This research addresses the sensor placement optimization problem (SPOP) in the industrial sector, aiming to enhance operational efficiency and safety through the strategic deployment of sensors. The focus is on optimizing the locations of thermal cameras and motion sensors, with a dual objective of maximizing coverage and minimizing redundancy in the production hall. To solve this challenge, a method based on the application of the nature-inspired bat algorithm was employed. The study reveals noteworthy findings, emphasizing the proficiency of the bat algorithm (BA) in optimizing the placement of thermal cameras and motion sensors. Numeric outcomes demonstrate the algorithm’s effectiveness in maximizing machine coverage while minimizing sensor usage within a real-world industrial environment. These results underscore the versatility and reliability of the BA, establishing it as a valuable tool for addressing complex optimization tasks in industrial settings.

Genetic Multi-Objective Optimization of Sensor Placement for SHM of Composite Structures

Increasingly often, due to the high sensitivity level of diagnostic systems, they are also sensitive to the occurrence of a significant number of false alarms. In particular, in structural health monitoring (SHM), the problem of optimal sensor placement (OSP) is appearing due to the need to reach a balance between performance and cost of the diagnostic system. The applied approach of considering nondominated solutions allows for adaption of the system parameters to the user’s expectations, treating this optimization problem as multi-objective. For this purpose, the NSGA-II algorithm was selected for the determination of an optimal set of parameters in the OSP problem for the detection of delamination in composite structures. The objectives comprise minimization of type-I and type-II errors, and number of sensors to be placed. The advantage of the proposed approach is that it is based on experimental data from the healthy structure, whereas all cases with a presence of delamination were acquired from numerical experiments. This makes it possible to develop a customized SHM system for the arbitrary location of damage.

Quantum‐Based Combinatorial Optimization for Optimal Sensor Placement in Civil Structures

Over the last decade, concepts such as industry 4.0 and the Internet of Things (IoT) have contributed to the increase in the availability and affordability of sensing technology. In this context, structural health monitoring (SHM) arises as an especially interesting field to integrate and develop these new sensing capabilities, given the criticality of structural application for human life and the elevated costs of manual monitoring. Due to the scale of structural systems, one of the main challenges when designing a modern monitoring system is the optimal sensor placement (OSP) problem. The OSP problem is combinatorial in nature, making its exact solution infeasible in most practical cases, usually requiring the use of metaheuristic optimization techniques. While approaches such as genetic algorithms (GAs) have been able to produce significant results in many practical case studies, their ability to scale up to more complex structures is still an area of open research. This study proposes a novel quantum‐based combinatorial optimization approach to solve the OSP problem approximately, within the context of SHM. For this purpose, a quadratic unconstrained binary optimization (QUBO) model formulation is developed, taking as a starting point of the modal strain energy (MSE) of the structure. The framework is tested using numerical simulations of Warren truss bridges of varying scales. The results obtained using the proposed framework are compared against exhaustive search approaches to verify their performance. More importantly, a detailed discussion of the current limitations of the technology and the future paths of research in the area is presented to the reader.

Optimal resilient sensor placement problem for secure state estimation

In this paper, we consider a sensor placement problem for secure state estimation in an adversarial environment. Specifically, this paper deals with a system consisting of n agents where up to l sensor measurements are potentially compromised by malicious adversaries and tackles the problem of sensor placement such that the system state can be reconstructed even if the measurements are compromised. However, since it is costly to deploy sensors for all agents, we also aim to place sensors at the lowest possible cost. We call this problem the optimal resilient sensor placement problem (ORSPP) and show that this problem is coNP-hard. Therefore, it is, in general, impossible to compute the ORSPP in polynomial time unless P=coNP. On the positive side, however, we also provide that this problem can be solved in polynomial time if the network structure of the agents satisfies certain conditions. The concrete algorithm for the ORSPP is provided, and further, in the process of providing the coNP-hardness, we also derive a new form of a necessary and sufficient condition for secure state estimation. Finally, we show the results of numerical simulations using a random graph and a diffusion process.

Optimal sensor placement for corrosion induced thickness loss monitoring in ship structures

Current monitoring strategies revolve around proactive maintenance. Corrosion-induced thickness loss (CITL) is one of the most common types of structural deterioration in ship hulls. Recently, Structural Health Monitoring (SHM) has emerged as an appealing alternative that may drive the transition to predictive maintenance. The complexity of ship structures requires that any SHM system must be designed to fully exploit information collected from a limited number of sensors. The aim of this work is to take a step in this direction by framing the design of an SHM system for CITL monitoring in hull structures under the guise of an optimization problem. The problem of detecting the existence of a certain level of thickness loss in hull structural elements is considered and treated using detection theory. The Optimal Sensor Placement (OSP) problem aims to generate strain sensor architectures that maximize the performance of the employed detector and minimize the costs incurred by its erroneous decisions. A computational implementation is considered, and synthetic strain data are generated using a detailed FE model of a reference vessel. Uncertainty is propagated to the data through a Monte Carlo Simulation (MCS). A Genetic Algorithm (GA) is employed to treat the optimization problem and the resultant sensor designs are compared to manually selected alternatives.

Iterative optimal sensor placement for adaptive structural identification using mobile sensors: Numerical application to a footbridge

This paper proposes an iterative optimal sensor placement (OSP) framework for structural identification and model updating of structural systems using a small number of mobile sensors. The model updating is performed through a Bayesian inference approach which is solved through asymptotic approximation for computational efficiency. In an iterative manner, the OSP is performed to minimize the information entropy in estimating the updating parameters of the model. Each OSP iteration is performed to find the next location of mobile sensors, where the prior probability distribution of updating parameters is assumed as the posterior probability distribution obtained from the previous iteration. This process is repeated until the uncertainties of updating parameters fall below a predetermined threshold. A forward sequential sensor placement algorithm is used to solve the OSP problem at each iteration. This algorithm provides a nearly-optimal solution and is much more efficient compared to an exhaustive search. This proposed framework is applied to a numerical case study, namely the Dowling Hall Footbridge located at Tufts University campus. Updating parameters used in this study are the added mass at different segments of the bridge deck. The purpose of using added mass is to create a realistic pseudo damage on a specific portion of the bridge. The proposed iterative system identification approach is applied for estimation of updating parameters considering different number of available sensors. This study shows that the iterative OSP approach using a small number of mobile sensors placed iteratively provides better model updating results compared to the case of using an optimal static sensor configuration involving larger number of sensors.

Two-timescale stochastic gradient descent in continuous time with applications to joint online parameter estimation and optimal sensor placement

In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely, the problem of joint online parameter estimation and optimal sensor placement for a partially observed diffusion process. We demonstrate how this can be formulated as a bilevel optimisation problem, and propose a solution in the form of a continuous-time, two-timescale, stochastic gradient descent algorithm. Furthermore, under suitable conditions on the latent signal, the filter, and the filter derivatives, we establish almost sure convergence of the online parameter estimates and optimal sensor placements to the stationary points of the asymptotic log-likelihood and asymptotic filter covariance, respectively. We also provide numerical examples, illustrating the application of the proposed methodology to a partially observed Beneš equation, and a partially observed stochastic advection-diffusion equation.

Optimal Geometries for AOA Localization in the Bayesian Sense

This paper considers the optimal sensor placement problem for angle-of-arrival (AOA) target localization in the 2D plane with a Gaussian prior. Optimal sensor locations are analytically determined for a single AOA sensor using the D- and A-optimality criteria and an approximation of the Bayesian Fisher information matrix (BFIM). Optimal sensor placement is shown to align with the minor axis of the prior covariance error ellipse for both optimality criteria. The approximate BFIM is argued to be valid for a sufficiently small prior covariance compared with the target range. Optimal sensor placement results obtained for Bayesian target localization are extended to manoeuvring target tracking. For sensor trajectory optimization subject to turn-rate constraints, numerical search methods based on the D- and A-optimality criteria as well as a new closed-form projection algorithm that aims to achieve alignment with the minor axis of the prior error ellipse are proposed. It is observed that the two optimality criteria generate significantly different optimal sensor trajectories despite having the same optimal sensor placement for the localization of a stationary target. Analysis results and the performance of the sensor trajectory optimization methods are demonstrated with simulation examples. It is observed that the new closed-form projection algorithm achieves superior tracking performance compared with the two numerical search methods.

Data-driven optimal sensor placement for high-dimensional system using annealing machine

We propose a novel method for solving optimal sensor placement problem for high-dimensional system using an annealing machine. The sensor points are calculated as a maximum clique problem of the graph, the edge weight of which is determined by the proper orthogonal decomposition mode obtained from data based on the fact that a high-dimensional system usually has a low-dimensional representation. Since the maximum clique problem is equivalent to the independent set problem of the complement graph, the independent set problem is solved using Fujitsu Digital Annealer. In contrast to existing greedy methods, which select the optimal point at each step and never reconsider the point selected previously, the proposed method is superior because it is able to find the optimal set of points. As a demonstration of high dimensional system, the pressure distribution measured by the pressure-sensitive paint method, which is an optical flow diagnose method, is reconstructed from the pressure data at the calculated sensor points. The root mean square errors (RMSEs) between the pressures measured by pressure transducers and the pressures reconstructed from the proposed method, an existing greedy method, and random selection method are compared. The similar RMSE is achieved by the proposed method using approximately 1/5 number of sensor points calculated by the existing method. This method is of great importance as a novel approach for optimal sensor placement problem and a new engineering application of an annealing machine.

Multi-objective sensor placement optimization of helicopter rotor blade based on Feature Selection

This work aims to develop a Structural Health Monitoring methodology that maximizes the acquired modal response and minimizes the number of sensors in a helicopter’s main rotor blade. Although this trade-off is a SHM principle, there is no methodology in literature that opposes these objectives for any structure. Firstly, a real AS350 helicopter rotor blade was experimentally tested and a numerical model was elaborated in FEM. An inverse method found the mechanical properties that fit numerical and experimental models. Then, a new methodology is proposed to address the Sensor Placement Optimization problem using the Multi-objective Lichtenberg Algorithm and Feature Selection. The Multi-objective Sensor Selection and Placement Optimization based on the Lichtenberg Algorithm (MOSSPOLA) has as one of the objectives the number of sensors and the other, one of the 7 best-known metrics in SPO: Kinetic Energy, Effective Independence, Average Driving-Point Residue, Eigenvalue Vector Product, Information Entropy, Fisher Information Matrix, and Modal Assurance Criterion. Pareto fronts and sensor configurations were generated and compared. Linear and convex families of Pareto fronts were unprecedentedly identified, showing a correlation between them. Better sensor distributions were associated with higher Hypervolume and the best metrics for each family were applied to damage identification for final comparison. The MOSSPOLA found a sensor configuration for each sensor number and metric, including one with 100% accuracy in identifying delamination considering triaxial modal displacements, minimum number of sensors, and noise for all blade sections.

Optimal Sensor Placement for Modal-Based Health Monitoring of a Composite Structure.

Optimal sensor placement is one of the important issues in monitoring the condition of structures, which has a major influence on monitoring system performance and cost. Due to this, it is still an open problem to find a compromise between these two parameters. In this study, the problem of optimal sensor placement was investigated for a composite plate with simulated internal damage. To solve this problem, different sensor placement methods with different constraint variants were applied. The advantage of the proposed approach is that information for sensor placement was used only from the structure’s healthy state. The results of the calculations according to sensor placement methods were subsets of possible sensor network candidates, which were evaluated using the aggregation of different metrics. The evaluation of selected sensor networks was performed and validated using machine learning techniques and visualized appropriately. Using the proposed approach, it was possible to precisely detect damage based on a limited number of strain sensors and mode shapes taken into consideration, which leads to efficient structural health monitoring with resource savings both in costs and computational time and complexity.

Optimal Arrangement of Structural Sensors in Landfill Based on Stress Wave Detection Technology

Sensor arrangement is the primary link of landfill structure health monitoring, and the number of measuring points and the quality of data directly affect the effect of modal identification. Therefore, how to ensure the service safety of landfill structure has become an important research content in the field of landfill structure health monitoring. In this paper, the stress wave propagation principle and stress wave detection theory are analyzed, and an optimal sensor arrangement method based on AGSA (adaptive gravity search algorithm) algorithm is proposed. Use CS (Cuckoo search) algorithm to optimize the objective function value. The corresponding response value is calculated by the finite element model, and the initial proxy model is constructed. By comparing and analyzing the results of model correction, it is found that the error of parameters before and after correction is less than 2.5%. It is further verified that AGSA algorithm can be used to solve the optimal sensor placement problem. In this paper, the use of structural health monitoring technology for health diagnosis and performance evaluation is an important means to ensure structural safety, prolong service life, and reduce maintenance cost. As the primary link of structural health monitoring, sensor system directly determines the accuracy of structural safety diagnosis.

A Single-Adversary-Single-Detector Zero-Sum Game in Networked Control Systems*

This paper proposes a game-theoretic approach to address the problem of optimal sensor placement for detecting cyber-attacks in networked control systems. The problem is formulated as a zero-sum game with two players, namely a malicious adversary and a detector. Given a protected target vertex, the detector places a sensor at a single vertex to monitor the system and detect the presence of the adversary. On the other hand, the adversary selects a single vertex through which to conduct a cyber-attack that maximally disrupts the target vertex while remaining undetected by the detector. As our first contribution, for a given pair of attack and monitor vertices and a known target vertex, the game payoff function is defined as the output-to-output gain of the respective system. Then, the paper characterizes the set of feasible actions by the detector that ensures bounded values of the game payoff. Finally, an algebraic sufficient condition is proposed to examine whether a given vertex belongs to the set of feasible monitor vertices. The optimal sensor placement is then determined by computing the mixed-strategy Nash equilibrium of the zero-sum game through linear programming. The approach is illustrated via a numerical example of a 10-vertex networked control system with a given target vertex.

Pareto-Based Bi-Objective Optimization Method of Sensor Placement in Structural Health Monitoring

For a practical structural health monitoring (SHM) system, the traditional single objective methods for optimal sensor placement (OSP) cannot always obtain the optimal result of sensor deployment without sacrificing other targets, which creates obstacles to the efficient use of the sensors. This study mainly focuses on establishing a bi-objective optimization method to select the sensor placement positions. The practical significance of several single-objective criteria for OSP is firstly discussed, based on which a novel bi-objective optimization method is proposed based on the Pareto optimization process, and the corresponding objective functions are established. Furthermore, the non-dominated sorting genetic algorithm is introduced to obtain a series of the Pareto optimal solutions, from which the final solution can be determined based on a new defined membership degree index. Finally, a numerical example of a plane truss is applied to illustrate the proposed method. The Pareto optimization-based bi-objective OSP framework presented in this study could be well suited for solving the problem of multi-objective OSP, which can effectively improve the efficiency of the limited sensors in SHM system.

Optimal sensor placement for identifying multi-component failures in a wind turbine gearbox using integrated condition monitoring scheme

Wind turbine gearbox has a high failure frequency and downtime, and therefore, several sensors are installed to perform condition monitoring to reduce the operation and maintenance costs. A gearbox can have infinite sensor nodal positions, but, in reality, the positioning of sensors is limited to a finite number of locations. Also, sensor location influences the quality of the data captured by the sensors, which is of key importance in a condition monitoring system. Hence selection of optimal sensor placement (OSP) is a challenging task which needs to be addressed. When the sensor type changes, the measurement response changes, and hence the OSP methodologies based on the measured responses may not work well. For addressing this, an optimization method based on statistical features is proposed to find the optimum sensor placement (OSP). In order to evaluate the effectiveness of the proposed method, experiments are conducted on a laboratory scale model of wind turbine gearbox considering multi-component faults and an integrated condition monitoring scheme. Variational mode decomposition and the Spearmen correlation coefficient are used to processes raw acoustic and vibration signals. Feature extraction is performed to obtain nine statistical features, and a mathematical objective function is constructed as a function of these features. Grey wolf optimizer is employed to find the optimal values of the statistical features. Fault classification is performed using Random forest (RF) and deep multi-layer perceptron (MLP) algorithms. Optimal sensor network identified by the above method reported classification accuracy of 86.88% and 88.34% for RF and MLP, respectively. As a result, the number of sensors reduced from eight to five. The proposed method can be used as an effective technique for OSP problems.

Optimal 3D Angle of Arrival Sensor Placement with Gaussian Priors.

Sensor placement is an important factor that may significantly affect the localization performance of a sensor network. This paper investigates the sensor placement optimization problem in three-dimensional (3D) space for angle of arrival (AOA) target localization with Gaussian priors. We first show that under the A-optimality criterion, the optimization problem can be transferred to be a diagonalizing process on the AOA-based Fisher information matrix (FIM). Secondly, we prove that the FIM follows the invariance property of the 3D rotation, and the Gaussian covariance matrix of the FIM can be diagonalized via 3D rotation. Based on this finding, an optimal sensor placement method using 3D rotation was created for when prior information exists as to the target location. Finally, several simulations were carried out to demonstrate the effectiveness of the proposed method. Compared with the existing methods, the mean squared error (MSE) of the maximum a posteriori (MAP) estimation using the proposed method is lower by at least when the number of sensors is between 3 and 6, while the estimation bias remains very close to zero (smaller than 0.15 m).

An enhanced sequential sensor optimization scheme and its application in the system identification of a rail-sleeper-ballast system

The problem of optimal sensor placement for system identification and damage detection is addressed by the development of a robust method based on Bayesian theory. Information entropy is used as the optimality measure to select the optimal configuration from the candidate configurations for a given number of sensors. The enhanced sequential sensor placement (ESSP) algorithm was developed to efficiently address the computational bottlenecks that may arise when a large number of measurable degrees of freedom (DOFs) are considered for candidate configurations. In this paper, the sensor redundancy problem in finely meshed models was addressed by considering (1) the spatial correction of prediction errors at measurable DOFs and (2) a minimum sensor interval. The proposed ESSP algorithm and the two strategies for handling sensor redundancy were studied by comparing the optimal configurations from the conventional methods and that from the ESSP algorithm for a rail-sleeper-ballast system. Finally, the optimal sensor configurations thus obtained were verified via model updating of an in-situ ballasted track system using measured data from an impact hammer test. The analysis results clearly show improvements in the optimality of the sensor configuration with the proposed method relative to the conventional methods.