No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Robust optimal sensor placement for operational modal analysis based on maximum expected utility
Abstract
Optimal sensor placement is essentially a decision problem under uncertainty. The maximum expected utility theory and a Bayesian linear model are used in this paper for robust sensor placement aimed at operational modal identification. To avoid nonlinear relations between modal parameters and measured responses, we choose to optimize the sensor locations relative to identifying modal responses. Since the modal responses contain all the information necessary to identify the modal parameters, the optimal sensor locations for modal response estimation provide at least a suboptimal solution for identification of modal parameters. First, a probabilistic model for sensor placement considering model uncertainty, load uncertainty and measurement error is proposed. The maximum expected utility theory is then applied with this model by considering utility functions based on three principles: quadratic loss, Shannon information, and K–L divergence. In addition, the prior covariance of modal responses under band-limited white-noise excitation is derived and the nearest Kronecker product approximation is employed to accelerate evaluation of the utility function. As demonstration and validation examples, sensor placements in a 16-degrees-of-freedom shear-type building and in Guangzhou TV Tower under ground motion and wind load are considered. Placements of individual displacement meter, velocimeter, accelerometer and placement of mixed sensors are illustrated.
... Furthermore, in Bayesian statistics, the FIM is often associated with an increase in expected Bayesian utility after making observations [5]. Bayesian utilities closely aligning with the FIM include the information entropy [6,7] and Kullback-Liebler (K-L) divergence [8]. ...
... The vertical axis corresponds to a scalarized index of the FIM, employed to facilitate the comparison between different number of sensors and between the four alternatives of EfI computation. The chosen scalar index, -ln(det(F)), is often interpreted as the information entropy, which quantifies the uncertainty involved in the parameter estimation [6,8]. A lower entropy, or a higher FIM determinant, indicates that the sensor arrangement has a higher informational value. ...
The concept of the Fisher information matrix (FIM) has been widely utilized for optimizing sensor placement on structures, often with the aim of maximizing detectability of modal changes and minimizing adverse effects from noise components. When using FIM for sensor placement optimization, the process is often guided by the effective independence (EfI) indices, which indicate the contribution of each sensor to the determinant of FIM. The EfI concept greatly facilitates the sensor placement process by identifying the least important sensor, which should be excluded from the candidate sensor set, as the one corresponding to the lowest EfI value. The basic expression for EfI is derived in conjunction with the simplest formulation of the FIM, where the identity matrix is assumed for the covariance of measurement errors. This study addresses the extension of the basic EfI to general FIMs that involve full error covariance matrices. Although the extension of the basic EfI has been attempted by previous studies, the existing approaches are shown to be lacking in rigor. Therefore, a rigorous expression for EfI is newly formulated in this study associated with full covariance matrix. The proposed EfI formula is not only accurate in capturing each sensor’s importance to the determinant of FIM, but also beneficial in improving computational efficiency. The mathematical rigor and computational significance of the proposed formula is confirmed through a case study using a continuous beam.
... (4) Te sensor method: Due to the limitations of processing technology, instruments, equipment, and measurement technology, the uncertainty in the structure to be monitored and the uncertainty in the data acquisition process are difcult to avoid and cannot be eliminated. To refect the infuence of the two factors on the optimal placement of sensors, extensive research has been conducted on the optimal placement of sensors in the feld of uncertainty such as sensitivity and robustness [161]. Castro-Triguero et al. [162] studied the optimal sensor layout scheme of the wooden structure by using the probability statistics method. ...
Active phased array antenna (APAA) is a representative of complex electronic equipment, and it plays significant roles in scenarios such as battlefield situation perception, aviation guidance, and communication. It has become the core equipment in land, navigation, and aeronautical applications. With the continuous improvement of technical changes and military requirements, the working frequency band, pointing accuracy, gain, and low sidelobe level of APAA increase, and the multi-disciplinary design contradiction between antenna electrical performance and structure and temperature becomes increasingly prominent. As a result, the electrical performance of APAA in service is prone to be affected by the external complex environments. The structural-electromagnetic-thermal (SET) coupling problem has become a key problem restricting the development of APAA. This paper has summarized the structural features and environmental loads of advanced APAA on different platforms and provided design basis and principle for antenna designer. And then the SET coupling theory of APAA has been introduced, which can be applied in both the design and manufacturing stage, as well as the performance control technology in service environment of APAA. This theory helps to analyze the impact of environmental factors, such as antenna structure deformation, radome high-temperature ablation, and feed errors, on the antenna's performance. For 128 × 768 spaceborne array antenna, in the range of 25∼85°C, the gain of antenna decreases with the increase of operating temperature and decreases by 0.015 dB with each increase of 1°C. The key design parameters in the fields of antenna manufacturing accuracy, efficient heat dissipation, and lightweight design are also analysed; for 32 × 32 rectangular planar phased array antenna, the gain of antenna decreases by 2.715 dB when the random error of installation position in x, y, and z direction reaches 1/10 of the wavelength. In addition, condition monitoring, displacement field reconstruction, and electrical performance compensation of APAA have also been touched to help engineers maintain and guarantee the antenna performance throughout its life cycle. Finally, the future research direction of SET technology of APAA has been discussed, and SET technology is extended to more fields such as antenna parameter uncertainty, high-frequency circuit electronics manufacturing, and electronic equipment performance guarantee.
... Information theoretic-based methods have also been used in structural dynamics to optimize the sensor configuration. In these methods a measure of the information gained from the sensor data, such as information entropy [53][54][55], mutual information [56,57], joint entropy [58], expected Kullback-Leibler divergence (KL-div) [59][60][61] and utility theory [62,63] computed from a Bayesian formulation is optimized with respect to the sensor locations. An OSP technique for response predictions based on Bayesian formulation and consistent with information theory has been developed in [64] in the case of known input. ...
An optimal sensor placement (OSP) framework for virtual sensing using the augmented Kalman Filter (AKF) technique is presented based on information and utility theory. The framework is applicable to input and response reconstruction when output-only vibration measurements are available. Using information theory, a utility function is built to quantify the expected information gained from the data for reducing the uncertainty of unmeasured quantities of interest (QoI). Taking into account the AKF equations and exploiting the Gaussian nature of the response QoI arising from linear dynamic systems, useful and informative analytical expressions are derived for the utility function. Subsequently, the utility function is extended to make the OSP design robust to uncertainties in the structural model and modelling error parameters, resulting in a multidimensional integral of the expected information gain over all possible values of the uncertain parameters, weighted by their assigned probability distributions. The proposed OSP framework maximizes this utility function through heuristic sequential sensor placement (SSP) strategies (forward and backward SSP) and genetic algorithms (GA) to optimize the type and location of sensors. A new modified SSP strategy is proposed that exploits the computational efficiency of the less accurate forward SSP algorithm and the accuracy of the computationally expensive backward SSP algorithm. A thorough study of the effect of measurements, modelling, input, and prediction uncertainties on the selection of the optimal sensor configuration is presented using a Finite Element (FE) model of a plate loaded by a single concentrated input force, highlighting the importance of accounting for robustness to errors and other uncertainties.
... In particular, techniques to avoid sensor clustering by taking into account the redundant information contained in the measurements have been discussed in [57,59]. OSP approaches based on expected KL-div and utility theory also address problems of parameter estimation of linear [70,[75][76][77] and nonlinear [78] structural systems, modal estimation [71,79] and structural health monitoring [80,81]. Finally, OSP issues related to multi-type sensor placement have also been addressed [56,65]. ...
A framework for optimal sensor placement (OSP) for virtual sensing using the modal expansion technique and taking into account uncertainties is presented based on information and utility theory. The framework is developed to handle virtual sensing under output-only vibration measurements. The OSP maximizes a utility function that quantifies the expected information gained from the data for reducing the uncertainty of quantities of interest (QoI) predicted at the virtual sensing locations. The utility function is extended to make the OSP design robust to uncertainties in structural model and modeling error parameters, resulting in a multidimensional integral of the expected information gain over all possible values of the uncertain parameters and weighted by their assigned probability distributions. Approximate methods are used to compute the multidimensional integral and solve the optimization problem that arises. The Gaussian nature of the response QoI is exploited to derive useful and informative analytical expressions for the utility function. A thorough study of the effect of model, prediction and measurement errors and their uncertainties, as well as the prior uncertainties in the modal coordinates on the selection of the optimal sensor configuration is presented, highlighting the importance of accounting for robustness to errors and other uncertainties.
... [70]. In Li and Der Kiureghian [85], three utility functions associated with quadratic loss, Shannon information and K-L divergence together with the elicitation of prior information are employed to make the prior decision on robust optimal sensor placement strategy. VoI analysis was not taken into account in their work due to the considered decision scenario. ...
The concept of Value of Information (VoI) has attracted significant attentions within the civil engineering community over especially the last decade. Triggered by the increasing focus on structural health monitoring, availability of data and emerging techniques of Big Data analysis and Artificial Intelligence, important insights on how to take benefit from VoI in structural integrity management have been gained. This literature review starts out with a summary of the historical developments and contains (1) a summary of two different VoI analysis origins, (2) a compilation of existing VoI analyses research and (3) current engineering interpretations and applications of VoI in the field of civil and infrastructure engineering. VoI analysis has roots in communication theory and Bayesian decision analysis in conjunction with utility theory. Starting point is thus taken in brief introduction of these theoretical foundations, followed by a discussion on the relevant modelling aspects such as information, probability and utility modelling. A detailed review of relevant existing research is presented, divided into the following main areas: computational methods, optimal sensor placement and engineering risk management. Finally, by way of conclusion and outlook, challenges and some promising directions for VoI analysis in the field of civil and infrastructure engineering are identified.
... It was demonstrated in an ultrasonic guided wave sensor design problem that Bayes risk can minimize the total presence of either type I or type II decision errors in SHM. The use of expected Kullback-Leibler (K-L) divergence or expected utility in sensor placement design [23,24,25] can also be classified as a type of Bayes risk. While the idea of minimizing expected risk (or maximizing the utility of your desired outcome) using Bayes risk is powerful for optimizing sensor placement under uncertainty, its advantageously generic nature and currently unexploited benefits must be carefully considered. ...
This paper presents a new approach to optimal sensor design for structural health monitoring (SHM) applications using a modified f-divergence objective functional. One of the primary goals of SHM is to infer the unknown and uncertain damage state parameter(s) from the acquired data or features derived from the data. In this work, we consider the loss of boundary contact (a ‘‘gap”) between a navigation lock miter gate and the supporting wall quoin block at the bottom of the gate to be the damage state parameter of concern. The design problem requires the optimal sensor placement of strain gages to obtain the best possible inference of the probability distribution of the gap length using the data from the multi-dimensional strain-gauge array. Using the notion of f divergences (measures of difference between probability distributions), a risk-adjustment is made by using functions that weigh the importance of acquiring useful information for a given true value of the state-parameter and using Bayesian optimization. For this case study of miter gate monitoring, a computationally expensive high-fidelity finite element model and its digital surrogate is employed to provide efficient, previously-validated data.
... To improve the quality of estimates when minimizing the number of measurement sensors, a sensor placement strategy described in, for example, [44][45][46][47][48], could be employed. These methods either require knowledge about the dynamic system in advance, for example by studying a finite element model, or require an iterative positioning process where sensors are moved around on the test structure in order to find the best combination. ...
The possibility of predicting the fatigue life of vehicle components based on sparse in-service vibration measurements enables an optimization of the maintenance and utilization of a vehicle fleet. This paper presents a novel methodology for fatigue damage prediction based on strain fields estimated from sparse vibration measurements on a dynamic structure together with a finite element model. The strain is estimated with a fixed-lag Kalman smoother, which is based on a state-space model derived from a modal description of the finite element model. The methodology is demonstrated with a full-scale laboratory experiment of a vehicle component subjected to road- induced vibrations where both strains and accelerations were measured. Fatigue damage is computed from the estimated strains, and verification of the accuracy of the predicted damage showed good results. The proposed methodology was then followed when the outcome of a full-scale vibration fatigue rig test of the component was predicted. The computed damage predicted the same failure locations as the ones which occurred during the rig test. The good correlation between the numerically computed damage and the outcome of the fatigue rig test indicates that the methodology presented in this paper has a high potential in predicting fatigue damage.
... By using the statistical method, Castro-Triguero et al. [22] paid attention to the uncertain parameters in material for placing sensors on a timber structure, and also examined the influence of uncertainties on sensor placement for a modal analysis on truss [23]. Based on a Bayesian linear model and maximum expected utility theory, Li et al. [24] demonstrated the robustness of sensor placement for operational modal identification. Yi et al. [25] investigated a novel multi-dimensional sensor placement method based on the triaxial EfI method, which can meet the demands of the application in Canton Tower. ...
... Because the number of sensors is limited, they must be placed in optimal positions for accurate identification. [4][5][6] For that reason, optimal sensor placement (OSP) has drawn the interest of many researchers over the last few decades. 7,8 Different methodologies have been proposed for OSP; most of them have utilized an iterative optimization algorithm to optimize one or more objective functions as the criteria to the suitability of sensors' location. ...
This paper presents a two‐stage optimal sensor placement method for modal identification of structures. At the first stage, using a graph theoretical technique, the structure is partitioned into equal substructures. At the second stage, a preset number of triaxial sensors are proportionately allocated to the substructures. The location of sensors is determined using an evolutionary optimization algorithm, which optimizes the triaxial modal assurance criterion of the structure. The first stage leads to the even distribution of the sensors. This stage not only improves the mode shape visualization as the secondary criterion but also accelerates the optimization process by space reduction. Here, various graph‐theoretical methods including k‐means, k‐means++, and spectral partitioning are examined as the partitioning techniques. In addition, a dynamic version for quantum‐inspired evolutionary optimization algorithm (DQEA) is proposed and applied to find the placement of triaxial sensors, along with the standard version of quantum‐inspired evolutionary algorithm and genetic algorithm. In order to examine the efficiency of the methods, the bridge model of the University of Central Florida, USA, is considered as the benchmark structure. The results show that the proposed method efficiently satisfies both criteria. Moreover, the introduced optimization algorithm (DQEA) outperforms other algorithms.
... The functional topology of the system, the physical model of the sensor information, the Bayesian inference technique and some constraints were used to determine the locations of sensors. Li et al. proposed the robust sensor placement for aimed at operational modal identification based on the maximum expected utility theory and a Bayesian linear model [26]. To avoid nonlinear relations between modal parameters and measured responses, sensor locations relative to identifying modal responses were also optimized. ...
Fault diagnosis usually requires lots of data which are collected through sensors mounted on some locations in the system. Performance of a diagnostic system is largely dependent upon the number and locations of sensors. Accordingly, optimization of sensor placement has a significant influence on the efficiency of fault diagnosis. In this paper, a novel sensor placement based on system reliability criterion is proposed, which aims to deal with the failure dependency and epistemic uncertainty. Specifically, it develops a dynamic fault tree (DFT) model to describe the dynamic failure behaviors based on failure mode and effects analysis and uses the interval numbers to express the failure rates of components. Furthermore, an indicator of sensor placement, named diagnostic importance factor (DIF), is calculated by mapping a DFT into a dynamic evidential network (DEN) and a sorting method based on the relative superiority degree is used to determine the potential locations according to DIF of components. In addition, the failure probability of the top event is considered as the criterion for sensor placement optimization and all scenarios of sensor placement are prioritized based on the criterion. Finally, the effectiveness of the proposed method is demonstrated via application to a real braking system.
... In order to have the most accurate and comprehensive understanding of the overall behavior of the studied building, a high number of sensors are usually located on the structure and their positioning is studied so as to maximize the measurable dynamic response [6,7], e.g. with the help of preliminary numerical estimates of the modal shapes by means of finite element models (FEMs) [8]. Sometimes, though, historic constructions are not completely reachable and only a limited number of positions are accessible or usable to install sensors. ...
The dynamic identification by ambient vibration data is widely used to supply information on the global health of structures through the investigation of changes in their modal parameters. It can be used even for verification of the state of damage of structures after hazardous threats, for example seismic activity. Therefore, it can play a crucial role to integrate and support conservation strategies for historic architectural assets. Sometimes, in historic constructions only a limited number of positions are accessible or usable to install sensors, and so modal analysis must be based on data from few measurement points. Moreover, they might not be the optimal positions for the studied structure, so the obtained results would need further verification. In such circumstances, the mutual validation between different modal analysis techniques can be useful to assess the reliability of results. In the present paper a case study of application to the so-called Temple of Minerva Medica, Rome, is described. Ambient vibration data were acquired in four rowing acquisition sessions carried out from July 2016 to July 2017, which is a timespan usable to assess the impact of the recent Central Italy seismic sequence. For problems related to the installation of the scaffolding only few points were available for instruments positioning. A variety of techniques were applied, including FRF, FDD, EFDD, SSI, HVSR and complex modal models. The variance of the modal parameters obtained by each different technique was utilized to provide indications on the reliability of the average values.
... The so called genetic algorithm is used to derive the performance-maximizing configuration. In Li et al. [22] a probabilistic model based on BN is developed that considers load uncertainty and measurement error. The optimal sensor placement is derived by optimizing three distinct utility functions that describe quadratic loss, Shannon information, and Kullback-Leibler divergence. ...
Oil pipeline network system health monitoring is important primarily due to the high cost of failure consequences. Optimal sensor selection helps provide more effective system health information from the perspective of economic and technical constraints. Optimization models confront different issues. For instance, many oil pipeline system performance models are inherently nonlinear, requiring nonlinear modelling. Optimization also confronts modeling uncertainties. Oil pipeline systems are among the most complicated and uncertain dynamic systems, as they include human elements, complex failure mechanisms, control systems, and most importantly component interactions. In this paper, an entropy-based Bayesian network optimization methodology for sensor selection and placement under uncertainty is developed. Entropy is a commonly used measure of information often been used to characterize uncertainty, particularly to quantify the effectiveness of measured signals of sensors in system health monitoring contexts. The entropy based Bayesian network optimization outlined herein also incorporates the effect that sensor reliability has on system information entropy content, which can also be related to the sensor cost. This approach is developed further by incorporating system information entropy and sensor costs in order to evaluate the performance of sensor combinations. The paper illustrates the approach using a simple oil pipeline network example. The so-called particle swarm optimization algorithm is used to solve the multi-objective optimization model, establishing the Pareto frontier.
... where R si k is measurement noise covariance matrix of ESM s i . H si k is the Jacobian matrix linearizing from the measurement function (7). ...
This paper researches the adaptive scheduling problem of multiple electronic support measures (multi-ESM) in a ground moving radar targets tracking application. It is a sequential decision-making problem in uncertain environment. For adaptive selection of appropriate ESMs, we generalize an approximate dynamic programming (ADP) framework to the dynamic case. We define the environment model and agent model, respectively. To handle the partially observable challenge, we apply the unsented Kalman filter (UKF) algorithm for belief state estimation. To reduce the computational burden, a simulation-based approach rollout with a redesigned base policy is proposed to approximate the long-term cumulative reward. Meanwhile, Monte Carlo sampling is combined into the rollout to estimate the expectation of the rewards. The experiments indicate that our method outperforms other strategies due to its better performance in larger-scale problems.
... Based on uncertain fault sensitivity, Blesa et al. [41] investigated the robustness of sensor locations. Using the Bayesian method and other statistical theories, Li and Kiureghian [42] investigated an ROSP approach to operate modal identification. Castro-Triguero et al. [43] studied the robustness and influence of uncertainty on OSP by considering different structural physical properties. ...
This paper proposes a robust optimal sensor placement method for structural health monitoring considering uncertainty. To avoid the deficiencies associated with scarce statistical information, a non-probabilistic approach is applied to cope with uncertainties in the optimal sensor placement field. Based on an interval analysis approach and a modal analysis method, an interval Fisher information matrix (IFIM) is derived from the deterministic case and the bounds of the IFIM eigenvalues are obtained. To realize the optimization process, the determinant of the IFIM, composed of the interval central and radius values corresponding to performance and fluctuation, is regarded as an optimization function. Following normalization and using weighted coefficients, the robust optimal sensor placement method with uncertain intervals can be transformed into a deterministic optimization process. Therefore, a single-objective optimization process can replace the two-objective optimization including the central and radius values. Because of the global optimization ability of modern intelligent algorithms, a genetic algorithm is adopted to determine the best sensor placement layout, with the node location used as the design variable. The validity of the proposed method is proved using three numerical examples.
... Li and Der Kiureghian in a recent study [9], develop a novel probabilistic framework for the OSP problem having as objective the maximization of the expected utility function from information theory. It is generic enough to accommodate any utility function, the authors though admit that there is no way to say which proposed configuration is the best unless specific evaluation criteria are introduced and posterior experimental evidence under various sensor configurations is obtained. ...
We revisit the optimal sensor placement of engineering structures problem with an
emphasis on in-plane dynamic strain measurements and to the direction of modal identification
as well as vibration-based damage detection for structural health monitoring
purposes. The approach utilized is based on the maximization of a norm of the Fisher
Information Matrix built with numerically obtained mode shapes of the structure and at
the same time prohibit the sensorization of neighbor degrees of freedom as well as those
carrying similar information, in order to obtain a satisfactory coverage. A new convergence
criterion of the Fisher Information Matrix (FIM) norm is proposed in order to deal with the
issue of choosing an appropriate sensor redundancy threshold, a concept recently introduced
but not further investigated concerning its choice. The sensor configurations obtained
via a forward sequential placement algorithm are sub-optimal in terms of FIM norm
values but the selected sensors are not allowed to be placed in neighbor degrees of
freedom providing thus a better coverage of the structure and a subsequent better identification
of the experimental mode shapes. The issue of how service induced damage
affects the initially nominated as optimal sensor configuration is also investigated and
reported. The numerical model of a composite sandwich panel serves as a representative
aerospace structure upon which our investigations are based.
... Li and Der Kiureghian in a recent study [9], develop a novel probabilistic framework for the OSP problem having as objective the maximization of the expected utility function from information theory. It is generic enough to accommodate any utility function, the authors though admit that there is no way to say which proposed configuration is the best unless specific evaluation criteria are introduced and posterior experimental evidence under various sensor configurations is obtained. ...
The optimal sensor placement of engineering structures problem with an emphasis on in-plane dynamic strain measurements and to the direction of modal identification as well as vibration-based damage detection for structural health monitoring purposes is revisited in this paper. The approach utilized is based on the maximization of a norm of the Fisher Information Matrix built with numerically obtained mode shapes of the structure and at the same time prohibit the sensorization of neighbor degrees of freedom as well as those carrying similar information, in order to obtain a satisfactory coverage. A new convergence criterion of the Fisher Information Matrix (FIM) norm is proposed in order to deal with the issue of choosing an appropriate sensor redundancy threshold, a concept recently introduced but not further investigated concerning its choice. The numerical model of a composite sandwich panel serves as a representative aerospace structure upon which our investigations are based.
... Installing fewer sensors at arbitrary locations also leads to either redundant or insufficient data that hamper the performance of fault diagnosis in the structures. To solve this problem, optimal sensor placement (OSP) strategy [2] is required that enables identifying sensor locations for efficient and economical data management. ...
Optimal sensor placement (OSP) plays a key role towards cost-effective structural health monitoring (SHM) of flexible structures like tall buildings. In the context of SHM, vibration measurements collected from a large array of sensors involve significant data storage, signal processing, and labor-intensive deployment of sensors. Moreover, with the increasing cost of vibration sensors, it is not possible to install sensors at all locations of the structure. To circumvent these practical challenges, the OSP provides a powerful mathematical framework to estimate unknown structural information based on optimally instrumented sensors located at fewer locations. The proposed research is focused on determination of optimum sensor positions in the multi-degrees-of-freedom system using blind source separation (BSS) method. The underdetermined signal separation capability of the BSS is explored to conduct the modal identification using fewer sensors and the resulting modal parameters are utilized to set up the optimization criterion for optimal sensor configurations. The methodology is illustrated using simulation models with different mass and stiffness distributions under a wide range of ground motion characteristics. Finally, the vibration measurements of the Canton tower data in China are utilized to demonstrate the performance of the proposed OSP technique.
... Especially information theory-based methods, in which the sensor locations are chosen to reduce the estimate uncertainties, have gained much attention. Typical approaches are to maximize some metric of the Fisher information matrix (FIM), see, for example, [2,3], or to minimize the information entropy, see, for example, [4,5]. Examination shows that the information entropy is linked to the determinant of the FIM, suggesting that minimization of the former is tantamount to maximization of the latter. ...
The present paper proposes an approach for deciding on sensor placements in the context of modal parameter estimation from vibration measurements. The approach is based on placing sensors, of which the amount is determined a priori, such that the minimum Fisher information that the frequency responses carry on the selected modal parameter subset is, in some sense, maximized. Or, more precisely, such that the largest value of the lowest attainable variance that can be realized by an unbiased estimator of the modal parameter set is minimized. The approach is validated in the context of a simple 10-DOF mass-spring-damper system by computing the variance of a set of identified modal parameters in a Monte Carlo setting for a set of sensor configurations, whose anticipated effectiveness are ranked according to the noted criterion. It is contended that the examined max-min criterion satisfies the objective of optimizing the accuracy of an identification setup more effectively than the more commonly used trace or determinant of the Fisher information matrix (FIM). It is shown that the widely used Effective Independence (EI) method, which uses the modal amplitudes as surrogates for the parameters of interest, provides sensor configurations yielding maximum variances that are up to 30 % larger than those obtained by use of the max-min approach.
Many optimal sensor placement (OSP) techniques have been developed basing on known external loads. However, it is often difficult to obtain excitation measurements. Therefore, the development of OSP under unknown inputs (OSP‐UI) is desirable. In this paper, based on modal Kalman filter (MKF), an OSP‐UI approach (MKF‐OSP‐UI) is proposed for optimally determining the number and locations of multitype sensors with the aim of minimizing the reconstructed responses errors. An MKF‐based approach previously developed by the authors is first employed for estimating multiscale structural responses and unknown loads. Then, an error covariance matrix is defined as a measure of the differences between the reconstructed responses and the corresponding actual ones. By using the covariance matrix of measurement noise for normalization, the ill‐conditioning problem caused by data fusion of multiscale responses is avoided. The sensors that have few contributions to the reconstructed responses are removed from the candidate set during iteration procedure. The sensor placement is finally determined when the estimation errors are below the preset level. Numerical results show that the sensor configuration determined by the proposed approach has a better performance on the joint estimation of multiscale responses and unknown inputs, as compared with that determined by experience.
A common approach to update a linear structural model with ambient vibration data or unmeasured excitation is to adopt a two-stage approach. The first stage involves identifying the modal parameters and the second stage involves updating the model parameters using the identified modal parameters. In this study, an optimal Bayesian sensor placement approach is proposed for such two-stage Bayesian model updating. The Bayesian sensor placement problem is formulated as an optimization problem in which the sensor configuration that maximizes the expected information gain in the model parameters is selected as the optimal one. Expressions to estimate the expected information gain in the model parameters are derived assuming a Gaussian posterior and small uncertainty in the model parameters. To illustrate the effectiveness of the proposed approach, two examples involving a simple 10-Degrees of Freedom (DOF) shear building model of a structure, and a 120-DOF space truss structure are considered. The effectiveness and applicability of the proposed approach is validated using the experimental data from a real 3-story frame structure.
Owing to an increased interest in damage identification and structural health monitoring during dynamic testing, sensor placement optimization has been a challenging research subject over the past decades. As the most influential method developed thus far, the effective independence (EI) method for sensor placement is discussed extensively in the present paper. Specifically, it was observed that, in the computation of the EI index, the sum of each retained row in the eigenvalue contribution matrix is a constant during each iteration. To reveal the significance of this property, a matrix transformation was employed, the form of which is similar to the orthogonal-triangular (QR) decomposition of the EI method previously proposed by the authors. In this regard, the physical significance of the re-orthogonalization of the modes through QR decomposition based on the computation of the EI method can be clearly seen from a new perspective. In view of this, a new sensor placement method, which is referred to as the remodeling innovation (RI) method, is proposed in this paper. The RI method avoids the existing problems in a conventional EI method in that, during each iterative calculation, the row with the least norm is removed without taking into account the effects of the structural mass redistribution. Furthermore, the RI method achieves better sensor placement in terms of the different evaluation criteria applied. Finally, to demonstrate the above observations, the proposed RI method was applied to the Hangzhou Nanshan Bridge.
Efficient Structural Health Monitoring (SHM) could reduce operation and maintenance costs, improve longevity, and enhance safety of the performance in complex mechanical systems. Traditional sensor network for SHM relied on simple sensor behavior without considering uncertain factors from system and environment. A probabilistic sensing model is required to simulate the realistic and stochastic sensor performance in the sensor network design process. In this paper, we introduce reliability-based design optimization for piezoelectric sensor network considering the detectability of different failure modes. The proposed method diagnoses failure based on the Mahalanobis Distance (MD) suitable for many SHM processes, while considering the uncertainties from structure properties and operation condition. Kriging is applied for surrogate modeling of the stochastic sensor signal and reduce computational cost. The optimal piezoelectric sensor network design is prototyped and its failure detection capability is experimentally verified.
The sensor placement methods for the modal identification are usually based on the prediction error between the measured and the structural model responses. Generally, the prediction errors, which are caused by the model error and the measurement noise, are assumed to be a Gaussian random vector together. This paper proposes a conditional information entropy based sensor placement method to separately investigate the influences of the measurement noise and the model error on the sensor placement. The changes in the element stiffness matrices are used to represent the model error. To quantify the uncertainty in the modal identification result, the conventional information entropy method is not suitable, due to the uncertainty in the Fisher matrix caused by the uncertain model error. This paper proposes a conditional information entropy index to quantify the uncertainty in the modal identification result with uncertain Fisher matrices. The optimal sensor placement corresponds to the one that maximizes the conditional information entropy index value over the set of all possible sensor placements. The conventional information entropy method can be seen as a special case of the proposed conditional information entropy method. In addition, the proposed sensor placement method can be applied to the placement of the multidimensional sensors. The simply supported beam and a bridge benchmark structure are used to explain the sensor placement method, and the redundancy threshold can be set to avoid redundant information provided by closely spaced sensors.
A Bayesian framework is presented for finding the optimal locations of strain sensors in a plate with a crack with the goal of identifying the crack properties, such as crack location, size, and orientation. Sensor grids of different type and size are considered. The Bayesian optimal sensor placement framework is rooted in information theory, and the optimal grid is the one which maximizes the expected information gain (Kullback–Liebler divergence) between the prior and posterior probability density functions of the crack parameters. The uncertainty in the crack parameters is accounted for naturally within the Bayesian framework through the prior probability density functions. The framework is demonstrated for a thin plate with crack, subjected to static loading. A finite element model is used to simulate the strain distributions in the plate given the crack properties. To verify the effectiveness of the proposed optimal sensor placement methodology, the estimated optimal sensor grids are used to perform Bayesian crack identification using simulated data. Parametric analyses are carried out giving emphasis on the effect of the number of sensors, grid type, and experimental data noise levels in the identification results.
Slender anti-air missiles experience longitudinal bending in supersonic flight and yet their autopilots are designed under the rigid-body assumption. Such autopilot design employs notch filters to remove the modal frequencies of the elastic airframe but this approach limits the autopilot bandwidth. In this paper, aeroservoelastic modelling and control of the ASTER 30 missile is proposed to enable autopilot design with extended bandwidth.
The aeroservoelastic model combines missile flight dynamics, actuator dynamics and airframe elasticity, the latter focusing on longitudinal bending treated as a continuous Euler–Bernoulli beam problem. The beam is discretised leading to a nodal model and the modal analysis is then performed. The modal model is expressed in the state-space form and its order is reduced to enable optimal sensor placement and active damping control. The aeroservoelastic model of the ASTER 30 missile is further refined for control purposes by optimally choosing actuator inputs together with the number and position of sensors to be mounted on the missile airframe. Once these choices are made, several variants of active vibration damping control are proposed and analysed in order to enable an extended bandwidth for the autopilot by countering the airframe deformation measured by these sensors.
Heuristic optimization algorithms have become a popular choice for solving complex and intricate sensor placement problems which are difficult to solve by traditional methods. This paper proposes a novel and interesting methodology called the asynchronous-climb monkey algorithm (AMA) for the optimum design of sensor arrays for a structural health monitoring system. Different from the existing algorithms, the dual-structure coding method is designed and adopted for the representation of the design variables. The asynchronous-climb process is incorporated in the proposed AMA that can adjust the trajectory of each individual dynamically in the search space according to its own experience and other monkeys. The concept of ‘monkey king’ is introduced in the AMA, which reflects the Darwinian principle of natural selection and can create an interaction network to correctly guide the movement of other monkeys. Numerical experiments are carried out using two different objective functions by considering the Canton Tower in China with or without the antenna mast to evaluate the performance of the proposed algorithm. Investigations have indicated that the proposed AMA exhibits faster convergence characteristics and can generate sensor configurations superior in all instances when compared to the conventional monkey algorithm. For structures with stiffness mutation such as the Canton Tower, the sensor placement needs to be considered for each part separately.
Operational modal analysis deals with the estimation of modal parameters from vibration data obtained in operational rather than laboratory conditions. This paper extensively reviews operational modal analysis approaches and related system identification methods. First, the mathematical models employed in identification are related to the equations of motion, and their modal structure is revealed. Then, strategies that are common to the vast majority of identification algorithms are discussed before detailing some powerful algorithms. The extraction and validation of modal parameter estimates and their uncertainties from the identified system models is discussed as well. Finally, different modal analysis approaches and algorithms are compared in an extensive Monte Carlo simulation study.
This paper provides a methodology for optimally locating sensors in a dynamic system so that data acquired from those locations will yield the best identification of the parameters to be identified. It addresses the following questions: (1) Given m sensors, where should they be placed in a spatially distributed dynamic system so that data from those locations will yield best estimates of the parameters that need to be identified?; and (2) given that we have already installed p sensors in a dynamic system, where should the next s be located? The methodology is rigorously founded on the Fisher information matrix and is applicable to both linear and nonlinear systems. A rapid algorithm is provided for use in large multi-degree-of-freedom systems. After developing the general methodology, the paper goes on to develop the method in detail for a linear N-degree-of-freedom, classically damped, system. Numerical examples are provided and it is verified that the optimal placement of sensors, as dictated by the methodology that is developed, could provide significantly improved estimates of the parameters to be identified.
A methodology is presented for designing cost-effective optimal sensor configurations for structural model updating and health monitoring purposes. The optimal sensor configurations ration is selected such that the resulting measured data are most informative about the condition of the structure. This selection is based on an information entropy measure of the uncertainty, in the model parameter estimates obtained using a statistical system identification method. The methodology is developed for the uncertain excitation case, encountered in practical applications for which data are to be taken either from ambient, vibration tests or from other uncertain excitations such as earthquake and wind. Important issues related to robustness of the optimal sensor configuration to uncertainties in the,, structural model are addressed. The theoretical developments are illustrated by designing the optimal configuration for a simple 8-DOF chain-like model of a structure subjected to an unmeasured base excitation and a 40-DOF truss model subjected to wind/earthquake excitation.
A method is presented for the selection of a set of sensor locations from a larger candidate set for the purpose of on-orbit identification and correlation of Large Space Structures. The method ranks the candidate sensor locations according to their contribution to the linear independence of the target modal partitions. In an iterative maner, the locations which do not contribute significantly are removed. The final sensor configuration tends to maximize determinant of the corresponding Fisher Information Matrix.
The Kronecker product has a rich and very pleasing algebra that supports a wide range of fast, elegant, and practical algorithms. Several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future. First, the application areas where Kronecker products abound are all thriving. These include signal processing, image processing, semidefinite programming, and quantum computing. Second, sparse factorizations and Kronecker products are proving to be a very effective way to look at fast linear transforms. Researchers have taken the Kronecker methodology as developed for the fast Fourier transform and used it to build exciting alternatives. Third, as computers get more powerful, researchers are more willing to entertain problems of high dimension and this leads to Kronecker products whenever low-dimension techniques are “tensored” together.
Engine health monitoring has been an area of intensive research for many years. Numerous methods
have been developed with the goal of determining a faithful picture of the engine condition. On the
other hand, the issue of sensor selection allowing an efficient diagnosis has received less attention
from the community. The present contribution revisits the problem of sensor selection for engine
performance monitoring within the scope of information theory. To this end, a metric that integrates
the essential elements of the sensor selection problem is defined from the Fisher information matrix. An example application consisting in a commercial turbofan engine illustrates the enhancement that can be expected from a wise selection of the sensor set.
The normative procedure for the design of an experiment is to select a utility function, assess the probabilities, and to choose that design of maximum expected utility. One difficulty with this view is that a scientist typically does not have, nor can be normally expected to have, a clear idea of the utility of his results. An alternative is to design an experiment to maximize the expected information to be gained from it. In this paper we show that the latter view is a special case of the former with an appropriate choice of the decision space and a reasonable constraint on the utility function. In particular, the Shannon concept of information is seen to play a more important role in experimental design than was hitherto thought possible.
A novel criterion of optimal sensor placement (OSP) for damage identification is proposed within rigorous mathematical framework. K-L divergence is employed to measure the performance of different configurations. The K-L divergence criterion aims at minimizing the probability distance of identified damage between by using all potential measuring points and by utilizing partial ones. An asymptotic approximation of distribution of damage parameters and expectation of K-L divergence are applied successively to make it feasible. It is shown that the K-L divergence is a versatile criterion which corresponds to the criteria of Fisher information, information entropy and Mahalanobis distance in its different types. In addition, an improved version of cross-entropy (CE) method is introduced to solve this binary combinatorial optimization problem. Finally, OSP problems of shear-type building and Aizhai suspension bridge are used to illustrate the efficiency of the K-L divergence criterion and the CE method.
A novel sensor placement method is proposed for structural health monitoring. The aim of the method is to select sensor locations from a set of possible candidate positions for achieving the best identification of modal frequencies and mode shapes. The proposed method depends on both the characteristics and the actual loading situations of a structure. It selects the sensor positions with the best subspace approximation of the vibration responses from the linear space spanned by the mode shapes. We search the best subspace approximation with a relative small subset through our recently proposed representative least squares method to approximate the original least squares estimator with the whole data set. The selected data subset is required to be representative and sufficiently approaching to portrait the scenario defined by the original full data set. The basic ideas of the representative least squares method are illustrated by a simple linear regression example. The algorithm for finding the representative subset for subspace approximation is described, and is applied to the I-40 Bridge.
Bayesian methods are a powerful tool in many areas of science and engineering, especially statistical physics, medical sciences, electrical engineering, and information sciences. They are also ideal for civil engineering applications, given the numerous types of modeling and parametric uncertainty in civil engineering problems. For example, earthquake ground motion cannot be predetermined at the structural design stage. Complete wind pressure profiles are difficult to measure under operating conditions. Material properties can be difficult to determine to a very precise level – especially concrete, rock, and soil. For air quality prediction, it is difficult to measure the hourly/daily pollutants generated by cars and factories within the area of concern. It is also difficult to obtain the updated air quality information of the surrounding cities. Furthermore, the meteorological conditions of day for prediction are also uncertain. These are just some of the civil engineering examples to which Bayesian probabilistic methods are applicable.
Familiarizes readers with the latest developments in the field
Includes identification problems for both dynamic and static systems
Addresses challenging civil engineering problems such as modal/model updating
Presents methods applicable to mechanical and aerospace engineering
Gives engineers and engineering students a concrete sense of implementation
Covers real-world case studies in civil engineering and beyond, such as:
structural health monitoring
seismic attenuation
finite-element model updating
hydraulic jump
artificial neural network
air quality prediction
Includes other insightful daily-life examples
Companion website with MATLAB code downloads for independent practice
Written by a leading expert in the use of Bayesian methods for civil engineering problems
This book is ideal for researchers and graduate students in civil and mechanical engineering or applied probability and statistics. Practicing engineers interested in the application of statistical methods to solve engineering problems will also find this to be a valuable text.
In the context of finite element model updating, experimentally obtained features are used to improve the quality of an initial finite element model. Using vibration tests, features like natural frequency, mode shapes, and modal damping ratios can be extracted from measured data. One possibility to perform such tests is a roving setup configuration that requires defining the positions of reference sensors to merge the information of all setups. Therefore, the determination of reference sensor positions is very important for reliable results.
The presented research is concentrated on the determination of optimal reference sensor positions assuming random excitations within a weakly stationary process. Predicted power spectral amplitudes and an initial finite element model are the basis to define the validation criterion of possible sensor positions. In combination with geometrically based design variables, which define the sensor positions, a genetic algorithm is applied to avoid the assessment of all possible combinations of reference sensor positions.
The applicability of the proposed approach is demonstrated on a numerical benchmark study of a simply supported beam and a case study of a real test specimen. Furthermore, the theory of determining the expected power spectral amplitudes is compared with results of vibration tests. It can be concluded that the proposed approach is suitable to determine optimal reference sensor positions as long as the initial finite element model has a sufficient accuracy.
A structural health monitoring system consists of permanently installed
sensors to collect structural information, and these sensors are
required to be placed at 'good' positions for damage identification.
Conventional sensor placement methods make use of dynamic
characteristics of a structure, i.e., mode shapes and natural
frequencies, to determine optimal sensor positions. However, these
methods do not take into account actual loading conditions and
structural responses. In fact, participation degree of interested mode
shapes in structural responses is of importance in both sensor placement
and subsequent modal identification. In the work, a novel load dependent
sensor placement method is developed by taking into consideration both
structural dynamic characteristics and actual loading conditions. The
objective of the proposed method is to achieve a nearly global unbiased
estimate of modal coordinates, and consequently best modal and damage
identification. It selects optimal sensor positions by subspace
approximation of actual structural responses with the linear space
spanned by interested mode shapes. Experiments are conducted on a
six-story truss structure to validate the proposed load dependent sensor
placement method. It is found that changing load conditions have to be
accounted for when the issue of sensor placement is seriously examined.
Furthermore, experiments have shown that better mode shape
identification can be achieved at sensor positions chosen by the
proposed novel load dependent sensor placement method than by
conventional techniques.
A modal approach is considered for sensor placement evaluation in
operational modal analysis (OMA) where modal participation at individual
degree of freedom (DOF) is evaluated separately for the target modes and
subsequently locations are identified using these participation
profiles. Modal contribution in output energy (MCOE) is proposed as
modal measure to evaluate modal participation and has been applied in
this modal approach framework for sensor placement evaluation. MCOE is
evaluated using observability grammian for any types of response
measurement (displacement, velocity or acceleration), while a system is
released from any initial condition. Further, existing modal measures
e.g. modal Hankel singular value (MHSV) and system norms (H2,
H∞ and Hankel) are explained in perspective of OMA. To understand
the efficiency of this proposed technique, MCOE is compared in terms of
modal participation with existing modal measures as well as with other
techniques like effective independence (EI) and modal kinetic energy
(MKE). Analytical similarity is found for participation of a mode with
EI method. Further, an existing large truss bridge structure is
considered for comparative study based on modal participation of
individual target modes along each DOF with acceleration measurement. In
this comparison, MCOE technique is found to be in very good agreement
with EI method as expected, while good agreement is observed with MHSV
as well as norms and reasonable agreement with MKE method. Further, the
adopted modal approach uses a flexible and insightful methodology for
sensor location evaluation for multiple target modes.
Closed-form solutions are presented for random vibration response integrals arising in the analysis of multi-degree-of-freedom (MDOF) systems to stationary nodal and/or support excitations Any pair of excitations must either be fully coherent (i.e., have identical frequency distribution) or totally incoherent. Fully coherent excitations may propagate with constant velocity, and have local amplitude variation. Solutions are presented for the response spectral moments under commonly used excitation spectra, including white noise, band-limited white noise, rational spectra, and spectra that are piecewise linear in log-log scale. These solutions provide complete generalizations of existing solutions, can save a great deal of computational effort in the random vibration analysis of large systems, and avoid difficulties that may be encountered in numerical integration when the integrands are highly oscillatory due to slow propagation velocities. It should be noted, however, that the solutions presented cannot be applied when the excitations are partially coherent.
Two methods are presented for structural sensor placement The first scheme selects the most linearly independent impulse responses at all candidate sensor locations from a Gram-Schmidt orthogonalization procedure. The second scheme is based on a principal component analysis and iteratively removes sensors that do not contribute significant information to the Fisher information matrix. Furthermore, use of a model reduction criterion is proposed to address the optimality issue. Several sensor placement methods were implemented to compare results and were applied to an Euler-Bernoulli beam and a cantilevered frame structure. It is shown that the proposed frequency criterion appears to be a selective criterion for choosing optimum sensor locations. Finally, the optimum measurement locations from several of the methods studied yield acceptable results based on data from an experimental study on the frame structure.
Uncertainties need to be taken into account for credible predictions of the dynamic response of complex structural systems in the high and medium frequency ranges of vibration. Such uncertainties should include uncertainties in the system parameters and those arising due to the modeling of a complex system. For most practical systems, the detailed and complete information regarding these two types of uncertainties is not available. In this paper, the Wishart random matrix model is proposed to quantify the total uncertainty in the mass, stiffness, and damping matrices when such detailed information regarding uncertainty is unavailable. Using two approaches, namely, (a) the maximum entropy approach; and (b) a matrix factorization approach, it is shown that the Wishart random matrix model is the simplest possible random matrix model for uncertainty quantification in discrete linear dynamical systems. Four possible approaches for identifying the parameters of the Wishart distribution are proposed and compared. It is shown that out of the four parameter choices, the best approach is when the mean of the inverse of the random matrices is same as the inverse of the mean of the corresponding matrix. A simple simulation algorithm is developed to implement the Wishart random matrix model in conjunction with the conventional finite-element method. The method is applied vibration of a cantilever plate with two different types of uncertainties across the frequency range. Statistics of dynamic responses obtained using the suggested Wishart random matrix model agree well with the results obtained from the direct Monte Carlo simulation.
A predictive decision-theoretic approach is developed for the Bayesian design problem. The loss functions used are fair Bayes, or proper scoring rules, and are quadratic measures of distance between probability measures. Optimal Bayesian designs are those which minimize the preposterior risk for the decision problem. Such designs typically depend on both the prior distribution and the loss function. The results are applied to certain normal regression models where explicit optimal designs are constructed.
A wide variety of model-based optimal test design methodologies have been developed in the past decade using deterministic approaches. This means that the test planning is based on a single-nominal model and an optimal design is obtained for precisely this model. Needless to say, the deterministic approach can lead to an ineffective distribution of sensors and poorly defined excitation points due to the presence of epistemic modelling errors. In this article, a robust-satisficing design approach to test planning is proposed based on info-gap decision theory. This methodology provides a decision-making tool for better understanding the trade-off between an optimal test design with no robustness to modelling uncertainties and a sub-optimal design which satisfies a less demanding level of performance while remaining maximally robust with respect to a given horizon of info-gap model uncertainty. The proposed strategy is illustrated using an aerospace application under base excitation conditions.
"This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times, tthe American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.
An iterative method, called effective independence, is used to place a small number of sensors on a large space structure for identification of a set of dynamically important mode shapes. The work is important in the area of on-orbit modal identification where a very limited number of sensors will be available. The method ranks sensor locations based upon their contribution to the linear independence of the target modal partitions. Linear independence must be maintained to perform test-analysis correlation to determine the accuracy of the analytical model with respect to the test data. The derived sensor configuration maintains the spatial independence of the target modes and maximizes the determinant of the corresponding Fisher information matrix. The proposed sensor placement strategy also minimizes the size of the test data matrix used by a popular modal identification technique called the eigensystem realization algorithm. A simple numerical example demonstrates the theory. The eigensystem realization algorithm is used to extract mode shapes, frequencies, and damping from simulated response data for varying levels of noise and damping. In all cases considered, the effective independence sensor configuration accurately identified more target modes than sensor sets derived using other sensor placement techniques.
The problem for measurement point selection in damage detection procedures is addressed. The concept of average mutual information is applied in order to find the optimal distance between measurement points. The idea is to select the measurement points in such a way that the taken measurements are independent, i.e. the measurements do not `learn' from each other. The average mutual information can be utilized as a kind of an autocorrelation function for the purpose. It gives the average amount of information that two points `learn' from each other. Thus the minimum of the average mutual information will provide the distance between measurement points with independent measurements. The idea to use the first minimum of the average mutual information is taken from nonlinear dynamics. The proposed approach is demonstrated on a test case. The results show that it is possible to decrease significantly the number of measurement points, without decreasing the precision of the solution.
Careful selection and placement of sensors are the critical issue in the construction and implementation of an effective structural health monitoring system. A hybrid method termed the optimal sensor placement strategy (OSPS) based on multiple optimization methods is proposed in this paper. The initial sensor placement is firstly obtained by the QR factorization. Then, using the minimization of the off-diagonal elements in the modal assurance criterion matrix as a measure of the utility of a sensor configuration, the quantity of the sensors is determined by the forward and backward sequential sensor placement algorithm together. Finally, the locations of the sensor are determined by the dual-structure coding-based generalized genetic algorithm (GGA). Taking the scientific calculation software matlab (MathWorks, Natick, MA, USA) as a platform, an OSPS toolbox, which is working as a black box, is developed based on the command-line compiling and graphical user interface-aided graphical interface design. The characteristic and operation method of the toolbox are introduced in detail, and the scheme selection of the OSP is carried out on the world's tallest TV tower (Guangzhou New TV Tower) based on the developed toolbox. The results indicate that the proposed method is effective and the software package has a friendly interface, plenty of functions, good expansibility and is easy to operate, which can be easily applied in practical engineering. Copyright © 2011 John Wiley & Sons, Ltd.
A criterion is proposed for making decisions regarding the optimal location of a given number of sensors to record the seismic response of a structure for identification purposes. The optimal location of the sensors is selected so that the expected value of a Bayesian loss function, expressed in terms of the Fisher information in the recordings, is minimized. The criterion is applied to the case of multi-degree-of-freedom systems with uncertain structural properties subjected to earthquake ground motions modelled as stationary stochastic processes. The use and capabilities of the criterion are thoroughly illustrated by means of an example. Results are used to assess the influence of record duration, recording noise, and ground motion frequency content and amplitude, on the optimal location of accelerometers as well as on the reduction of prior uncertainty about the structural parameters. © 1998 John Wiley & Sons, Ltd.
The Guangzhou New TV Tower (GNTVT), currently being constructed in Guangzhou, China, is a supertall structure with a height of 610 m. This tube-in-tube structure comprises a reinforced concrete inner tube and a steel outer tube adopting concrete-filled-tube columns. A sophisticated structural health monitoring (SHM) system consisting of over 600 sensors has been designed and is being implemented by The Hong Kong Polytechnic University to GNTVT for both in-construction and in-service real-time monitoring. This paper outlines the technology innovation in developing and implementing this SHM system, which includes (i) modular design of the SHM system, (ii) integration of the in-construction monitoring system and the in-service monitoring system, (iii) wireless-based data acquisition and Internet-based remote data transmission, (iv) design and implementation of a fiber Bragg grating sensing system,(v) structural health and condition assessment using static and dynamic monitoring data, (vi) verification of the effectiveness of vibration control devices by the SHM system, and (vii) development of an SHM benchmark problem by taking GNTVT as a test bed and using real-world measurement data. Preliminary monitoring data including those obtained during the Wenchuan earthquake and recent typhoons are also presented. Copyright © 2008 John Wiley & Sons, Ltd.
In the work presented here we aim to demonstrate the utility of the observability matrix condition number as a metric in developing optimum measurement strategies in a range of process engineering contexts via some new examples. To demonstrate use of the approach itself we consider controller tuning in the field to show how response parameters can be estimated with a minimum of carefully timed empirical measurements. In front-end process development also we show how timing of the minimal number of measurements needed for chemical kinetic parameter estimation can be decided, with potential savings in empirical laboratory resource overheads. In the context of equipment specification, we show how capital investment can be balanced against the quality of data delivered for plant operation with an example in selecting the number and location of temperature sensors in a batch distillation. Finally, constructing the observability matrix itself can be problematic, especially in non-linear applications where we propose, as a feasible approach, its direct calculation from simulated measurement responses and their sensitivity to the unknown state and parameters.
Statistical modeling of space-time data has often been based on separable covariance functions, that is, covariances that can be written as a product of a purely spatial covariance and a purely temporal covariance. The main reason is that the structure of separable covariances dramatically reduces the number of parameters in the covariance matrix and thus facilitates computational procedures for large space-time data sets. In this paper, we discuss separable approximations of nonseparable space-time covariance matrices. Specifically, we describe the nearest Kronecker product approximation, in the Frobenius norm, of a space-time covariance matrix. The algorithm is simple to implement and the solution preserves properties of the space-time covariance matrix, such as symmetry, positive definiteness, and other structures. The separable approximation allows for fast kriging of large space-time data sets. We present several illustrative examples based on an application to data of Irish wind speeds, showing that only small differences in prediction error arise while computational savings for large data sets can be obtained. Copyright © 2007 John Wiley & Sons, Ltd.
We propose two block preconditioners for Toeplitz-block matrices (i.e. each block is Toeplitz), intended to be used in conjunction with conjugate gradient methods. These preconditioners employ and extend existing circulant preconditioners for point Toeplitz matrices. The two preconditioners differ in whether the point circulant approximation is used once or twice, and also in the cost per step. We discuss efficient implementation of these two preconditioners, as well as some basic theoretical properties (such as preservation of symmetry and positive definiteness). We report results of numerical experiments, including an example from active noise control, to compare their performance.
In this paper we address the problem of Bayesian experimental design and estimation for ANOVA models when the prior distribution belongs to a class of finite mixtures of normal distributions. We introduce new optimality criteria and derive an estimator and the corresponding experimental design which simultaneously are optimal for the class. The proposed estimator minimizes an average posterior expected loss over the class of posterior distributions, the experimental design minimizes an average of the Bayes risks of the proposed estimator over this class. The optimization of this criterion is mathematically tractable, allowing us to give closed form solutions for the estimation problem even in more complex ANOVA models such as two-way ANOVA and block designs. For the one-way ANOVA model we also derive an alternative estimator and the corresponding design according to a minimax-regret criterion. The similar performance of these two procedures is illustrated in an example.
In this paper we discuss the choice of sensor positions for a tubular reactor with a preset number of sensors. Different observability measures, based on the observability matrix (Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, NJ, 1980; Callier and Desoer, Linear System Theory, Springer-Verlag, Berlin, 1991: Damak et al., DYCORD '92, 1992, pp. 315–320), the observability gramian (Callier and Desoer, 1991) as well as on the Popov-Belevitch-Hautus rank test (Kailath. 1980) will be considered for locating optimal sensor positions. The analysis is carried out on the reduced finite-dimensional model of the process. The results of these investigations will be illustrated in simulation and put in perspective with the modal observability properties of the original infinite-dimensional model.
Theoretical and computational issues arising in the selection of the optimal sensor configuration for parameter estimation in structural dynamics are addressed. The information entropy, measuring the uncertainty in the system parameters, is used as the performance measure of a sensor configuration. A useful asymptotic approximation for the information entropy, valid for a large number of measured data, is derived. The asymptotic estimate is then used to rigorously justify that selections of the optimal sensor configuration can be based solely on a nominal structural model, ignoring the time history details of the measured data which are not available in the experimental design stage. It is further shown that the lower and upper bounds of the information entropy are decreasing functions of the number of sensors. Based on this result, two algorithms are proposed for constructing effective sensor configurations that are superior, in terms of computational efficiency and accuracy, to the sensor configurations provided by genetic algorithms. The theoretical developments and the effectiveness of the proposed algorithms are illustrated by designing the optimal configuration for a 10-degree-of-freedom (d.o.f.) chain-like spring–mass model and a 240-d.o.f. three-dimensional truss structure.
This paper introduces a novel approach for optimal sensor and/or actuator placement for structural health monitoring (SHM) applications. Starting from a general formulation of Bayes risk, we derive a global optimality criterion within a detection theory framework. The optimal configuration is then established as the one that minimizes the expected total presence of either type I or type II error during the damage detection process. While the approach is suitable for many sensing/actuation SHM processes, we focus on the example of active sensing using guided ultrasonic waves by implementing an appropriate statistical model of the wave propagation and feature extraction process. This example implements both pulse-echo and pitch-catch actuation schemes and takes into account line-of-site visibility and non-uniform damage probabilities over the monitored structure. The optimization space is searched using a genetic algorithm with a time-varying mutation rate. We provide three actuator/sensor placement test problems and discuss the optimal solutions generated by the algorithm.
We address the problem of finding a design that minimizes the Bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior. Uncertainty in the prior is formulated in terms of having a family of priors instead of one single prior. Two different classes of priors are considered: is a family of conjugate priors, and a second family of priors is induced by a metric on the space of nonnegative measures. The family has earlier been suggested by Leamer and Polasek, while was considered by DeRobertis and Hartigan and Berger. The setup assumed is that of a canonical normal linear model with independent homoscedastic errors. Optimal robust designs are considered for the problem of estimating the vector of regression coefficients or a linear combination of the regression coefficients and also for testing and set estimation problems. Concrete examples are given for polynomial regression and completely randomized designs. A very surprising finding is that for , the same design is optimal for a variety of different problems with different loss structures. In general, the results for are significantly more substantive. Our results are applicable to group decision making and reconciliation of opinions among experts with different priors.
Elicitation is a key task for subjectivist Bayesians. Although skeptics hold that elicitation cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subject-matter expert colleagues. This article reviews the state of the art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful; that is, what criteria should be used. Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily "true" in some objectivistic sense, and cannot be judged in that way. We see that elicitation as simply part of the process of statistical modeling. Indeed, in a hierarchical model at which point the likelihood ends and the prior begins is ambiguous. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions. The psychological literature suggests that people are prone to certain heuristics and biases in how they respond to situations involving uncertainty. As a result, some of the ways of asking questions about uncertain quantities are preferable to others, and appear to be more reliable. However, data are lacking on exactly how well the various methods work, because it is unclear, other than by asking using an elicitation method, just what the person believes. Consequently, one is reduced to indirect means of assessing elicitation methods. The tool chest of methods is growing. Historically, the first methods involved choosing hyperparameters using conjugate prior families, at a time when these were the only families for which posterior distributions could be computed. Modern computational methods, such as Markov chain Monte Carlo, have freed elicitation from this constraint. As a result, now both parametric and nonparametric methods are available for low-dimensional problems. High-dimensional problems are probably best thought of as lacking another hierarchical level, which has the effect of reducing the as-yet-unelicited parameter space. Special considerations apply to the elicitation of group opinions. Informal methods, such as Delphi, encourage the participants to discuss the issue in the hope of reaching consensus. Formal methods, such as weighted averages or logarithmic opinion pools, each have mathematical characteristics that are uncomfortable. Finally, there is the question of what a group opinion even means, because it is not necessarily the opinion of any participant.
Sensor network optimization using Bayesian networks, decision graphs, and value of information
- C Marlings
- M Pozzi
C. Marlings, M. Pozzi, Sensor network optimization using Bayesian networks, decision graphs, and value of information, in: Proceeding of 12th
International Conference on Applications of Statistics and Probability in Civil Engineering, Vancouver, Canada, 2015.
Optimal sensor placement for structural health monitoring based on K–L divergence, in: Proceeding of Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures
- B Li
- J Ou
B. Li, J. Ou, Optimal sensor placement for structural health monitoring based on K–L divergence, in: Proceeding of Safety, Reliability, Risk and Life-Cycle
Performance of Structures and Infrastructures, New York, USA, 2013.
Information Theoretic Optimal Sensor Placement in Structural Health Monitoring (Master thesis)
- B Li
B. Li, Information Theoretic Optimal Sensor Placement in Structural Health Monitoring (Master thesis), Dalian University of Technology, Dalian, China,
2012.
Automatic choice of measurement locations for modal survey test
- J Penny
- M Friswell
- S Garvey
J. Penny, M. Friswell, S. Garvey, Automatic choice of measurement locations for modal survey test, AIAA J. 32 (1994) 407-414.
- W Heylen
- S Lammens
- P Sas
W. Heylen, S. Lammens, P. Sas, Modal Analysis Theory and Testing, Katholieke Unversiteit Leuven, Leuven, Belgium, 1998.