Clustering of Sensor Locations Using the Effective Independence

AIAA JOURNAL Vol. 53, No. 5, May 2015

Technical Notes Clustering of Sensor Locations Using the Effective Independence Method

II.

Michael I. Friswell∗ Swansea University, Swansea SA2 8PP, United Kingdom and Rafael Castro-Triguero† University of Cordoba, 14071 Cordoba, Spain DOI: 10.2514/1.J053503

I.

Effective Independence Method

The EFI method, developed by Kammer [10], is one of the most popular methods to optimally locate sensors in structural dynamic tests. The starting point of the EFI method is the full modal matrix from a finite-element model of the structure to be studied. The finiteelement model could be built with any type of elements, but not all DOFs under consideration can be measured. Some of these DOFs are internal, while others are rotations, neither of which can be measured conveniently. Hence, the rows corresponding to DOFs that cannot be measured are deleted from the full modal matrix. Furthermore, system identification methodologies from sensor data records can only extract some of the mode shapes of the structure under study. Therefore, a limited number of target modes must be selected in order to find the best sensor configuration. Consequently, only some rows (potential DOF locations) and columns (target modes) of the full modal matrix are retained. The EFI sensor placement algorithm is based on the FIM F, which is defined as

Introduction

T

HE optimal placement of a limited number of sensors in a structure is usually performed to obtain accurate experimental real-time data. In particular, sensor placement is considered one of the most important pretest steps in experimental/operational modal analysis [1–4]. Structural health monitoring techniques also incorporate optimal sensor placement (OSP) methodologies for different purposes: system identification, structural damage identification, and finite-element updating [5–9]. Classical OSP approaches are suboptimal methods. Kammer [10] proposed an iterative method, the effective independence method (EFI), based on the maximization of the determinant of the Fisher information matrix (FIM). This matrix is defined as the product of the mode shape matrix and its transpose. The number of sensors is reduced in an iterative way by deleting the degrees of freedom (DOFs) successively from the mode shape matrix. This idea can also be extended to biaxial and triaxial accelerometers [11]. The FIM can also be weighted by the use of the mass matrix from a finite-element model [12]. Energy matrix rank optimization techniques are similar to the EFI method, but in this case, the strain energy of the structure is maximized rather than the determinant of the FIM [13]. A similar method based on the kinetic energy is obtained by using the mass matrix instead of the stiffness matrix [14]. Information entropy has also been used as the performance measure of a sensor configuration, as information entropy is directly related to parametric uncertainty. Papadimitriou and Beck [15,16] developed an entropy-based optimal sensor location method for model updating in structural dynamics. In structural dynamics applications, the mode shapes are often very significant, and the ability to distinguish between the modes is important. Hence, the selection of the sensor locations should ensure that the mode shapes are independent. Many methods, such as the EFI method, are often assumed to optimize the independence of the modes, although in general this is not true. This note investigates the relationship between the EFI method and the linear independence of the modes.

F  Φ⊤ Φ

(1)

where Φ is the mode shape matrix. The FIM is symmetric and positive semidefinite. Furthermore, if the mode shape vectors are linearly independent, then the FIM is full rank; i.e., the rank is equal to the number of target mode shapes. The main aim of the EFI method is to select the best DOF configuration (in which sensors are to be placed) that maximizes the FIM determinant. These DOFs are selected in an iterative way, thereby producing a suboptimal solution to the problem. The selection procedure is based on the orthogonal projection matrix E, defined as E  ΦF−1 Φ⊤

(2)

The matrix E is an idempotent matrix of which the rank is equal to the sum of the diagonal terms. Hence, the ith diagonal element of matrix E, denoted by Eii, represents the fractional contribution of the ith DOF to the rank of E. The DOF with the lowest value of Eii is deleted as a candidate sensor location, and the corresponding row is removed from the modal matrix. The contribution of Eii to the rank of E is often cited as the reason that this procedure retains the linear independence of the mode shapes. However, it is easily shown [9] that detF  detF0 1 − Eii 

(3)

where F0 is the original FIM and F is the FIM after the removal of the ith sensor. Thus, every time one row is deleted from the mode shape matrix Φ, the determinant of the FIM decreases, and the objective of the selection method is to maintain a high value for this determinant. Many authors (for example, those of [17]) have highlighted that the high determinant ensures the variance of the estimated response is low, and this is used as the criteria for sensor location selection. However, this selection process does not necessarily optimize the independence of the mode shape matrix, as is often quoted in the OSP literature. Here, the independence of the mode shapes is quantified by the condition number of the mode shape matrix. At the extremes, the linear independence of the modes and the determinant of the FIM have well-defined values. For example, if the modes are linearly dependent, then the condition number will be infinity, and the determinant of the FIM will be zero. In contrast, if the modes are orthonormal, then the condition number is unity, and the determinant of the FIM is equal to the number of modes considered. For intermediate cases, there is no direct relationship between the

Received 8 April 2014; revision received 19 August 2014; accepted for publication 6 January 2015; published online 12 March 2015. Copyright © 2014 by Michael I Friswell and Rafael Castro-Triguero. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-385X/ 15 and $10.00 in correspondence with the CCC. *Professor of Aerospace Structures, College of Engineering; m.i.friswell@ swansea.ac.uk. † Department of Mechanics, Campus de Rabanales. 1388

AIAA JOURNAL, VOL. 53, NO. 5:

TECHNICAL NOTES

1389

determinant and linear independence. This may be understood by considering the singular value decomposition of the FIM; the determinant is the product of the singular values, whereas the condition number of the mode shape matrix is the square root of the ratio of the highest to lowest singular values.

III.

EFI and Independence of Modes

Lateral Displacement

A cantilever beam model is used to demonstrate the properties of the EFI sensor selection method and the effect on the determinant of the FIM and the independence of the modes. By focusing on a very simple structure, the properties of the method are highlighted and emphasized. A finite-element model of the cantilever beam is constructed with 30 elements. The beam has a length of 0.45 m, a width of 20 mm, and a thickness of 2 mm, although these dimensions are arbitrary for the purposes of the OSP demonstration. The translational DOF at every node of the finite-element mesh of the beam is a candidate sensor location. Only the first three mode shapes of the cantilever beam, shown in Fig. 1, are considered as the target modes. The modes are mass normalized before the rows corresponding to the rotational DOFs are removed. Figure 2 shows the results of the sensor selection using the EFI method. When the number of sensors equals the number of modes (three in this case) the locations selected are well distributed on the beam. However, when more sensors than modes of interest are selected, then the sensors cluster in groups, in which the number of groups is the same as the number of modes. This is clearly shown in Fig. 2, in which more sensors are selected, and this phenomenon has been verified for a range of numbers of modes and sensors in the beam example. The locations for seven and eight sensors are not given but are consistent with those shown in Fig. 2. This clustering also occurs in many other papers, particularly for beam and plate structures (see, for example, [18,19]). For smooth mode shapes, the measurements within a cluster are likely to be highly correlated and hence will represent a poor choice of locations, particularly when large numbers of sensors are selected. Furthermore, the finite-element model used to choose the sensor locations will inevitably contain errors, and so in practice, the sensors will not be placed perfectly; using well-spaced sensor locations will improve the robustness to modeling errors. For comparison, an exhaustive search is performed to select three, four, and five sensor locations for this example, by evaluating every possible combination of sensor locations. The objective is to choose the sensors that give the minimum condition number and hence give the highest level of independence of the mode shapes. Large condition numbers indicate a poor choice of sensor configuration. A condition number of 1 indicates that the mode shape matrix is orthogonal, which is very unlikely even when all of the DOFs are measured because the mode shapes are orthogonal with respect to the mass matrix. Figure 3 shows the results of this search, which gives a much more distributed set of sensor locations. The choice of the five sensor locations looks slightly strange, in that locations 11 and 12 are

Fig. 2 Sensor locations selected by the EFI method for three target modes and nine, six, five, four, and three sensors.

chosen; however, inspection of the mode shapes in Fig. 1 shows that the slopes of the second and third mode shapes are relatively large with opposite signs in this region, and hence the mode shapes are highly sensitive to the measurement location. Note also that the same sensor locations are not chosen as the number of sensors increases; any method to choose sensor locations that relies on adding or removing sensor locations one at a time is likely to perform poorly. The locations chosen by the EFI method and the exhaustive search are given in Table 1, together with the determinant of the FIM in each case and the condition number of the mode shape matrix. Also shown are the results when all 30 sensors are retained. The EFI method does meet its objective of maximizing the determinant of the FIM (indeed for three and four sensors, an exhaustive search shows the EFI sensor selection gives the maximum determinant). However, the condition number of the mode shape matrix is significantly higher than for the sensors chosen by an exhaustive search. Table 1 also shows that the condition number of the mode shape matrix when all of the DOFs are measured is greater than the condition number obtained by the exhaustive search for four and five sensors. This does not imply that using four or five sensors is better than using 30 sensors but occurs because the mode shapes are orthonormal with respect to the mass matrix. One possibility is to consider M0.5 Φ rather than Φ in the exhaustive search, where M is the mass matrix; the corresponding vectors are orthogonal when all

Distance Along Beam

Fig. 1 The first three mode shapes of the cantilever beam, clamped at the left end. Only the neutral axis of the beam is shown, for vibration in a single plane in the thickness direction. The circles represent the candidate sensor locations.

Fig. 3 Sensor locations selected by an exhaustive search for the minimum condition number for three target modes and five, four, and three sensors.

1390

AIAA JOURNAL, VOL. 53, NO. 5:

Table 1 Sensor selection for the EFI method, with the FIM determinant and the condition number of the mode shape matrix; results of an exhaustive search to minimize the condition number of the mode shape matrix also shown Method All sensors EFI

Number of sensors 30 9

EFI

6

EFI Exhaustive EFI Exhaustive EFI Exhaustive

5 5 4 4 3 3

Locations selected

detFIM

condΦ

1, 2, : : : , 29, 30 10, 11, 12, 20, 21, 22, 28, 29, 30 10, 11, 20, 21, 22, 29, 30 10, 11, 20, 21, 30 11, 12, 19, 22, 28 10, 11, 21, 30 7, 14, 20, 27 11, 21, 30 10, 19, 27

1.181 × 107 9.996 × 105

1.1002 1.7113

3.435 × 105

1.8742

1.938 × 105 9.840 × 104 9.776 × 104 2.902 × 104 4.888 × 104 1.892 × 104

1.5155 1.0433 1.9983 1.0906 2.0139 1.1358

DOFs are considered, and initial results do suggest that the condition number does then reduce with the number of sensors chosen. However, this approach is not pursued further, as the emphasis of this Note is the clustering of the sensors in the EFI method, and the condition number of the mode shape matrix using the DOFs chosen by the EFI method is always significantly higher than the condition number for the DOFs chosen by the exhaustive search.

IV.

Conclusions

This Note has considered the optimal placement of sensors in structural dynamics and has clearly demonstrated that the EFI method maximizes the determinant of the FIM but does not maximize the linear independence of the modes, evaluated by the condition number. When more sensors are selected than modes of interest, the EFI method tends to produce clusters of sensor locations, which do not appear to be ideal from an engineering perspective. However, when the number of sensors equals the number of modes, the EFI method does give a set of sensors that are distributed throughout the structure. An exhaustive search based the condition number of the mode shape matrix shows that there are significantly better sensor locations, if the objective is directly related to the mode shape independence. Furthermore, these sets of sensors are more evenly distributed throughout the structure. Mode shape independence is highlighted as a desirable objective in this Note, and this has been demonstrated by an exhaustive search with a small number of sensors on a simple structure. Even for a small number of sensors, the combinatorial explosion of the number of sensor sets to explore makes an exhaustive search infeasible for practical structures. The challenge is to develop an efficient iterative selection method that minimizes the condition number. Initial studies have shown that starting with a full set of sensor locations and removing locations one by one produces a poor set of sensor locations. Starting with a small set of sensors (perhaps obtained by an exhaustive search) and adding sensors seems to be a more promising approach. Furthermore, when a sensor is either added or removed, there appears to be no fast and convenient method to assess the contribution of each candidate degree of freedom to the condition number; each location has to be added or removed in turn, and the condition number has to be calculated. This is the subject of ongoing research.

References [1] Pickrel, C. R., “A Practical Approach to Modal Pretest Design,” Mechanical Systems and Signal Processing, Vol. 13, No. 2, 1999, pp. 271–295. doi:10.1006/mssp.1998.1212 [2] Penny, J. E. T., Friswell, M. I., and Garvey, S. D., “Automatic Choice of Measurement Locations for Dynamic Testing,” AIAA Journal, Vol. 32, No. 2, 1994, pp. 407–414. doi:10.2514/3.11998

TECHNICAL NOTES

[3] Papadopoulos, M., and Garcia, E., “Sensor Placement Methodologies for Dynamic Testing,” AIAA Journal, Vol. 36, No. 2, 1998, pp. 256–263. doi:10.2514/2.7509 [4] Garvey, S. D., Friswell, M. I., and Penny, J. E. T., “Evaluation of a Method for Automatic Selection of Measurement Locations Based on Subspace-Matching,” XIV International Modal Analysis Conference (IMAC), Society of Experimental Mechanics, Bethel, CT, Feb. 1996, pp. 1546–1552. [5] He, L., Lian, J., Ma, B., and Wang, H., “Optimal Multiaxial Sensor Placement for Modal Identification of Large Structures,” Structural Control and Health Monitoring, Vol. 21, No. 1, 2014, pp. 61–79. doi:10.1002/stc.1550 [6] D'Souza, K., and Epureanu, B., “Sensor Placement for Damage Detection in Nonlinear Systems Using System Augmentations,” 48th AIAA/American Soc. of Mechanical Engineers/American Helicopter Soc./American Society for Composites Structures, Structural Dynamics, and Materials Conference, AIAA Paper 2007-2057, 2007. [7] Reynier, M., and Kandil, H. A., “Sensors Location for Updating Problems,” Mechanical Systems and Signal Processing, Vol. 13, No. 2, 1999, pp. 297–314. doi:10.1006/mssp.1998.1213 [8] Castro-Triguero, R., Murugan, S. M., Gallego, R., and Friswell, M. I., “Robust Sensor Placement under Parametric Uncertainty,” Mechanical Systems and Signal Processing, Vol. 41, Nos. 1–2, Dec. 2013, pp. 268– 287. doi:10.1016/j.ymssp.2013.06.022 [9] Castro-Triguero, R., Saavedra Flores, E. I., Diaz de la O, F. A., Friswell, M. I., and Gallego, R., “Optimal Sensor Placement in Timber Structures by Means of a Multi-Scale Approach with Material Uncertainty,” Structural Control and Health Monitoring, Vol. 21, No. 12, Dec. 2014, pp. 1437–1452. doi:10.1002/stc.1654 [10] Kammer, D. C., “Sensor Placement for On-Orbit Modal Identification and Correlation of Large Space Structures,” Journal of Guidance, Control and Dynamics, Vol. 14, No. 2, 1991, pp. 251–259. doi:10.2514/3.20635 [11] Kammer, D. C., and Tinker, M. L., “Optimal Placement of Triaxial Accelerometers for Modal Vibration Tests,” Mechanical Systems and Signal Processing, Vol. 18, No. 1, 2004, pp. 29–41. doi:10.1016/S0888-3270(03)00017-7 [12] Kammer, D. C., and Peck, J. A., “Mass-Weighting Methods for Sensor Placement Using Sensor Set Expansion Techniques,” Mechanical Systems and Signal Processing, Vol. 22, No. 7, 2008, pp. 1515–1525. doi:10.1016/j.ymssp.2008.01.002 [13] Hemez, F. M., and Farhat, C., “An Energy Based Optimum Sensor Placement Criterion and Its Application to Structural Damage Detection,” XII International Modal Analysis Conference (IMAC), Society of Experimental Mechanics, Bethel, CT, 1994, pp. 1568–1575. [14] Heo, G., Wang, M. L., and Satpathi, D., “Optimal Transducer Placement for Health Monitoring of Long Span Bridge,” Soil Dynamics and Earthquake Engineering, Vol. 16, Nos. 7–8, 1997, pp. 495–502. doi:10.1016/S0267-7261(97)00010-9 [15] Papadimitriou, C., and Beck, J. L., “Entropy-Based Optimal Sensor Location for Structural Model Updating,” Journal of Vibration and Control, Vol. 6, No. 5, 2000, pp. 781–800. doi:10.1177/107754630000600508 [16] Papadimitriou, C., “Optimal Sensor Placement Methodology for Parametric Identification of Structural Systems,” Journal of Sound and Vibration, Vol. 278, Nos. 4–5, 2004, pp. 923–947. doi:10.1016/j.jsv.2003.10.063 [17] Li, D. S., Li, H. N., and Fritzen, C. P., “Load Dependent Sensor Placement Method: Theory and Experimental Validation,” Mechanical Systems and Signal Processing, Vol. 31, Aug. 2012, pp. 217–227. doi:10.1016/j.ymssp.2012.04.014 [18] Rao, A. R. M., and Anandakumar, G., “Optimal Sensor Placement Techniques for System Identification and Health Monitoring of Civil Structures,” Smart Structures and Systems, Vol. 4, No. 4, 2008, pp. 465– 492. doi:10.12989/sss.2008.4.4.465 [19] Chang, M., and Pakzad, S. N., “Optimal Sensor Placement for Modal Identification of Bridge Systems Considering Number of Sensing Nodes,” Journal of Bridge Engineering, Vol. 19, No. 6, 2014, Paper 04014019. doi:10.1061/(ASCE)BE.1943-5592.0000594

B. Epureanu Associate Editor