MODERN spacecraft [1], [2], offshore platforms [3], and

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1 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, Robust Optimal Sensor Placement for Uncertain Structures With Interval Parameters Chen Yang, Zixing Lu, and Zhenyu Yang Abstract 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. Index Terms Optimal sensor placement, structural health monitoring, interval robust optimization, Fisher information matrix, genetic algorithm. I. INTRODUCTION MODERN spacecraft [1], [2], offshore platforms [3], and high-rise buildings [4] are increasingly large-scale and complex. During their lifetimes, such structures may deteriorate because of the extreme service environments. Therefore, to maintain the structure in a safe condition, structural health monitoring (SHM) must be considered and applied. One of the most important components of an SHM system is the sensor subsystem [5], [6]. The optimal sensor placement (OSP) largely determines two inverse problems, damage identification [7] [9] and load identification [10] [12]. Recently, the use of advanced technology to select optimum sensor locations has attracted considerable attention [13] [22]. Manuscript received December 4, 2017; accepted December 30, Date of publication January 4, 2018; date of current version January 31, This work was supported in part by the National Natural Science Foundation of China under Grant , in part by the Beijing Natural Science Foundation under Grant , in part by the National Natural Science Foundation of China under Grant and Grant , in part by the Defense Industrial Technology Development Program under Grant JCKY B001 and Grant JCKY C001, and in part by the Fundamental Research Funds for the Central Universities under Grant YWF-16-BJ-Y-63. The associate editor coordinating the review of this paper and approving it for publication was Prof. Giancarlo Fortino. (Corresponding author: Chen Yang.) The authors are with the Institute of Solid Mechanics, Beihang University, Beijing , China ( cyang@buaa.edu.cn; luzixing@buaa.edu.cn; zyyang@buaa.edu.cn). Digital Object Identifier /JSEN It is well known that sensors must be placed at important locations to obtain the best dynamic information by using modal test technology for parameter identification, damage detection, response estimation, etc. To maximize the linear independence of the targeted modes, the effective independence (EfI) method [23] is widely used. Moreover, based on the modal kinetic energy (MKE), which plays an important role in structural dynamics, Udwadia [24] first compared the maximum energy from all possible locations to determine the sensor placement, rather than simply relying on experiences. In addition, the modal assurance criterion (MAC) index is a useful tool to describe modal shape correlations. Therefore, Carne and Dohrmann [25] proposed a method for reducing the complex iterative process in OSP problems by minimizing the MAC. Based on the basic ideas of classical methods, many advanced approaches have been investigated in the OSP field, including sequence selection methods and intelligent optimization algorithms. For instance, based on the Fisher information matrix, Yao et al. [26] proposed a modified genetic algorithm to select sensors for modal identification. Because of their global optimization ability, modern intelligent algorithms such as genetic algorithms (GAs) [27] [30], particle swarm optimization (PSO) [31], and simulated annealing algorithms (SAAs) [32], [33] can be applied to select the best sensor locations. Based on the Euclidean distance and statistical perturbation, Yi et al. [13] modified a monkey algorithm for OSP to increase the solution efficiency. Furthermore, an integer-coding format was applied to operate the design variables. According to the nearest neighbor index, Lian et al. [17] used a discrete PSO (DPSO) and investigated a novel objective function to avoid the imperfections of traditional EfIs. The feasibility and reliability of this OSP method have been demonstrated in actual projects. Based on the maximization of the fitness function, a custom GA for SHM in a stadium was investigated, and the layout of sensor locations was also determined [34]. Without obtaining the optimization gradient information, the studies mentioned above have shown the advantages of intelligent algorithms in the OSP field. Therefore, a GA is employed in this study to solve the robust OSP (ROSP) problem with interval parameters. In practice, uncertain models and noise in the sampling process are too significant to be neglected. Statistical methods are suitable for describing uncertainty and have been widely used in the OSP field [35] [38]. Moreover, a twostage stochastic optimization solution has been proposed to select the optimal hydrophone measurements for different noise levels. By combining probabilistic finite element IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 2032 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, 2018 analysis, damage identification, and optimization concepts, Guratzsch and Mahadevan [39] proposed a series of sensor methodologies. Recently, to reveal the influence of uncertainty on OSP, many researchers have studied the design, analysis, and optimization of robustness in sensor placement fields. Based on uncertainty restrictions, Carr et al. [40] constructed three robust optimization models with corresponding criteria and discussed the effects of changes to different parameters. 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. As a primary method for addressing the uncertainty problems, statistical approaches have been successfully applied in many fields. However, statistical methods have limitations with respect to quantifying the uncertainties of probabilistic distributions of structural models, responses, and measurements. First, the detailed probability density functions (PDFs) for the uncertainties are not always precise, especially for some advanced and long-lifetime structures, such as space solar power satellites [29]. Second, it is also inaccurate to use early-stage PDFs to describe the later stages of longlifetime structures such as the international space station (ISS), which has a service life of over 30 years. Moreover, it is difficult to obtain the PDFs of uncertainties in some cases. For instance, the uncertainties of structural sections and material properties can be attributed to a series of processes such as production and manufacturing. Fortunately, in the past two decades, the non-probabilistic approach [44], [45] has emerged as a method for tackling uncertainties, and can resolve this predicament. Recently, researchers have studied the non-probabilistic approach to inverse problems [7] in sensor [46] and actuator [47] placement which have numerous uncertainties. In our opinion, the non-probabilistic approach has significant benefits in solving uncertainty problems. Although this method has recently been applied to OSP [46], previous studies have only focused on the influence of interval uncertainties on deterministic sensor placement layouts. Specifically, Reference [46] only analyzed the reliability of current sensor locations under uncertainties. However, deterministic optimal layouts are not always the best locations in uncertain structures, and no method for obtaining the best sensor layout for an uncertain case was provided in [46]. Therefore, using conventional OSP methods to obtain placements under deterministic conditions may not achieve the best results in structures containing uncertainties. Moreover, because of the limitations of conventional statistical methods, several strong requirements for ROSP using interval analysis algorithms have been revealed. According to the discussion above, the shortcomings associated with current OSP methods applied to uncertain structures are clearly understood. This paper aims to fill the gap left by these methods and determine the best sensor locations by combining a non-probabilistic approach and robust optimization method. To avoid the deficiencies of scarce statistical information, a non-probabilistic approach is applied to cope with uncertainties in the OSP field. As long as we know the bounds of the uncertainties, an interval Fisher information matrix (IFIM) can be conveniently obtained by applying interval analysis technology to the deterministic case. The determinant of the IFIM consists of interval central and radius values corresponding to the performance and fluctuation of the OSP, respectively. To realize the optimization algorithm, this determinant is regarded as a fitness function. Therefore, the proposed ROSP method with uncertain intervals can be transformed into a deterministic optimization process by applying normalization and using the weighted coefficients. Thus, a single-objective optimization process replaces the twoobjective optimization including interval central and radius values. A GA is adopted to obtain the best sensor locations, and the validity of the ROSP method is verified using three numerical examples. This paper is organized as follows. A fitness function based on the Fisher information matrix (FIM) is derived in Sec. II. This fitness function is extended to the interval parameters in Sec. III, and ROSP using the interval analysis method is introduced in Sec. IV. To clearly describe the proposed algorithm, the ROSP solution using GA is derived in Sec. V. Numerical examples and discussions are presented in Sec. VI. Finally, we summarize the paper and draw conclusions in Sec. VII. II. FITNESS FUNCTION BASED ON FISHER INFORMATION MATRIX In deterministic OSP, as an evolution of EfI, the FIM is considered to be the fitness function. Because the FIM indicates the independence of modal shapes, the OSP problem can be addressed as follows: place m sensors within a structure to present the best fit for N modes [46]. The main aim of EfI [23] is to realize maximize the linear independence in the N modes. Based on the modal superposition algorithm, a response u can be obtained from the N modes as follows: u = q + ω = N ϕ i q i + ω (1) where is a modal matrix with n N dimensions; n is the number of candidates; N is the total number of orders of the sampled modes; and ϕ i is the i-th column in. ω and ω are the noise vector and element, respectively; q i and q are the generalized coordinate column and matrix, respectively. According to the above basic dynamic theory, the sensor placement process is expressed as follows: to maximize the linear independence and completeness of modal information, we must understand how to select m placements from n possible locations [46]. That is how to determine the best estimate of the true ˆq. Therefore, an unbiased estimator for ˆq is applied to estimate the errors: [ ] 1 1 J = E[(q ˆq)(q ˆq) T ]= σ 2 T = σ 2 [ Q ] 1 (2) where Q is the FIM and σ 2 is the variance. After simplifying the above statement, the assumptions regarding the noise i=1

3 YANG et al.: ROSP FOR UNCERTAIN STRUCTURES WITH INTERVAL PARAMETERS 2033 for all candidates are independent with the same statistical distribution. The best estimation of q is obtained when Q is maximized. The determinant is a key matrix property [17], and so the best sensor locations can be identified using the determinant of the FIM. Furthermore, the fitness function f in the deterministic case for OSP can be expressed as ( ) f = det T = det (Q) (3) Higher values of this fitness function indicate better sensor placement layouts. According to matrix theory, the determinant of Q can be transformed into a product of all eigenvalues: f = N λ i (4) i=1 where λ i is the i-th eigenvalue of Q. Based on the FIM mentioned above, a fitness function for OSP is proposed in the deterministic case. III. ROBUST OPTIMAL SENSOR PLACEMENT METHOD USING INTERVAL ANALYSIS A. Interval Modal Analysis Based on the modal analysis method and the interval analysis algorithm, Sim et al. [48] proposed a modal interval analysis approach for obtaining the ranges of modes. This method is adopted in this paper to estimate uncertain modes. The main procedure can be described as follows. Step 1. The bounds of interval numbers b j must be known. The deterministic modes ϕi c are obtained based on dynamic theory. Step 2. Based on first-order perturbation theory, the partial derivative of the modes, i (b c ) ϕ b j, can be obtained, where b c is the central value of the interval and b j denotes the j-th uncertain parameter. Step 3. After b j, ϕi c,and ϕ i (b c ) b j have been obtained, the interval bounds of modes are given by ϕ i = ϕi c j=1 ϕ i (b c ) b j b j and ϕ i = ϕi c + ϕ i (b c ) b b j. j=1 j Detailed derivations can be found in [48]. B. Interval Analysis of FIM When uncertainties exist in the stiffness and mass of a structure, its responses and characteristics could be different to the deterministic case. Thus, it may influence the FIM Q value. To avoid deficiencies associated with insufficient statistical information on uncertainties, uncertain structural parameters are considered to be non-probabilistic intervals, giving the IFIM Q I. According to the interval mathematics, the IFIM Q I can be written as: Q I = Q c + Q I (5) where Q c and Q I denote the central and radius IFIM values, respectively. Applying a first-order Taylor expansion, IFIM can be approximated as Q I (b) = Q ( b c) + Q ( b c) b I (6) b In addition, Eq. (6) can be rewritten in component form: Q I (b) = Q ( b c) un Q ( b c) + b I j (7) b j where un denotes the uncertain number. Applying interval extension theory to Eq. (7) gives Q I (b) = Q ( b c) un Q ( b c) + b j bi j (8) j=1 j=1 where b I j = b j [ 1, 1]. The bounds of IFIM are: Q (b) = Q ( b c) un Q ( b c) b j b j (9) j=1 Q (b) = Q ( b c) un Q ( b c) + b j b j (10) where the derivative Q(bc ) b j can be obtained from: Q ( b c) = Q ( s) s (ϕ i ) ϕ ( i b c ) (11) b j s ϕ i b j where s is the sub-matrix of, which corresponds to the degree of freedoms (DOFs) for sensor placement. Moreover, the following two equations are easily derived: Q ( s ) = 2 s (12) s s (ϕ i ) =[0,..., 1,..., 0 ] (13) ϕ i Hence, according to Eqs. (12) and (13), and based on the discussion above, the derivative Q(bc ) b j can be easily computed and the bounds of the IFIM can be obtained by Eqs. (9) and (10). C. Interval Fitness Function for ROSP According to Sec. II and Eq. (3), the determinant of the FIM can be regarded as an important property for OSP. Therefore, based on the interval analysis method, the interval determinant of IFIM can be written as: ( ) [ ( ) ] T f I = det Qs I = det s I I s (14) j=1 The interval fitness f I can be described as: f I = f c + f I = f c + f [ 1, 1] = [ ] f, f (15) The determinant interval for the IFIM can be solved using the product of the series of its eigenvalue intervals as: N f I = λi I (16) i=1

4 2034 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, 2018 where λ I i is the ith eigenvalue interval of I s. λi i can be described in terms of the interval center λ c i and the radius λ i, to form: λ I i = λ c i + λi i = λ c i + λ i [ 1, 1] (17) Furthermore, λ i can be obtained as follows [49]: λ i = ( ψ c i ) T Qs ψ c i (18) where ψ c i is the eigenvector of Q s corresponding to λ c i. Hence, λ i = λ c i λ i (19) λ i = λ c i + λ i (20) Furthermore, according to interval mathematics [49], the interval numbers of the fitness function can be expressed as: N f = λ i (21) f = i=1 N λ i (22) f = f f (23) 2 Therefore, the bounds of the IFIM determinant have been obtained. IV. ROSP WITH INTERVAL ANALYSIS METHOD The classical deterministic OSP can be extended into an uncertain ROSP problem. Based on previous research [17], the classical OSP in the deterministic case can be described as: i=1 max f (d) s.t. d d d (24) where d is a design variable denoting the sensor placements. d and d are the interval bounds of d, and f is the fitness function, which is the objective function of the OSP problem. In this paper, f is the determinant of the FIM. Conventional OSPs are based on the deterministic method, and are only suitable for structures without uncertainties. However, the uncertainties that exist in actual engineering projects cannot be neglected, and the influences of these uncertainties on OSP layout cannot be ignored. Furthermore, using the deterministic optimization method to deal with uncertain structures may result in errors. Therefore, the proposed ROSP is applied to solve the sensor placement problem with uncertainties, as shown in Fig. 1. When the OSP problem is deterministic (without any uncertainties), the black solid line denotes the deterministic design curve with the maximal fitness function signified by the orange point. This problem can be referred to as deterministic optimal sensor placement (DOSP). In addition, the fluctuations in the fitness function are shown as two curves (lower bound and upper bounds) when uncertainties exist. Although the value of the orange point is greater than that of the green point, the orange point exhibits larger fluctuations. Therefore, the orange point Fig. 1. Schematic diagram of robust optimization. may not be the best solution for uncertain cases. In other words, the design variable for the green point is not sensitive to the uncertainties, which indicates that it is a more robust solution. This robust optimization can be applied to obtain the robust optimal solution, which can be expressed by: max f (d, b) s.t. d d d b b I (25) As shown in Eq. (25), the fitness function contains the design variable d and uncertainty b. Robust optimization can be defined as follows: seek the solution that maximizes the fitness function while minimizing the fluctuations. Therefore, according to Eq. (25), robust optimization can be transformed into a multiple-objective optimization problem: max f ( d, b c) min f (d) s.t. d d d b b I (26) where f represents the fluctuations in the fitness function achieved from the uncertainty propagation analysis. The radius value caused by uncertainties is regarded as a fluctuation in the ROSP problem. Therefore, the lower bound of the fitness function is maximal. To seek the best ROSP solution, the two-objective optimization is replaced by a single-objective optimization according to the following normalization and weighted coefficients [47]: max f robust = α f c ( d, b c) f (d) f (1 α) f s.t. d d d b b I (27) where f is the optimal solution of the fitness function for a deterministic case. f is the fluctuation at f for an uncertain case, and α is a weighting factor that provides a trade-off between performance and fluctuation. V. ROSP SOLUTION USING GA As is well known, it is difficult to seek a global optimal solution using conventional gradient-based optimization algorithms. Because the partial derivatives of the fitness function (determinant of IFIM) with respect to the design variables

5 YANG et al.: ROSP FOR UNCERTAIN STRUCTURES WITH INTERVAL PARAMETERS 2035 Fig. 2. Flow chart of ROSP. (sensor placements) are difficult to compute, conventional optimization algorithms cannot be applied in the OSP field [47]. Fortunately, modern intelligent optimization algorithms can overcome this drawback. In the proposed method, a GA is applied to seek the best solution. GAs do not require gradients or partial derivatives. Based on Darwinian evolution, GAs distribute a fitness value to each candidate solution [50], and then apply the principle of survival of the fittest. The GA considers all possible solutions to be genetic, with new individuals is produced from existing individuals through selection, crossover, and mutation processes. This procedure of evolution approaches the best solution after sufficient number of generations. According to the basic concept, all candidate design variables should first be coded. Furthermore, FEM node numbers are assumed to be design variables. The IFIM is then obtained using interval analysis method. Furthermore, an ROSP fitness function must be constructed and the GA can be applied to obtain the best solution to the sensor placement problem. Detailed flow charts of the ROSP solution process and GA optimization process are shown in Fig. 2 and Fig. 3, respectively. Step 1 (Coding): All of the possible FEM node numbers should first be coded; these are regarded as the design variables in GA. Step 2 (Initialization): In the first iteration, the initial individuals should be randomly generated. Fig. 3. Flow chart of GA optimization process. Step 3 (IFIM): Based on the interval analysis method, a classical FIM is extended to an uncertain IFIM, as investigated in Sec. III-B. Step 4 (Determinant): Based on interval analysis and the IFIM obtained above, the interval bounds of the

6 2036 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, 2018 Fig. 4. Schematic diagram of 45-story shear building. IFIM determinant can be estimated, as demonstrated in Sec. III-C. Step 5 (Fitness): The ROSP fitness function, including performance and fluctuation, can be constituted as described in Sec. IV. Step 6 (Evaluation): The GA searches for the best fitness solution in each iteration. In addition, the algorithm evaluates the current individuals according to some convergence criterion. Step 7 (GA): The GA processes of selection, crossover, and mutation are applied. The individuals are updated in each iteration of the evolution process. Step 8 (Decoding): After the best individuals have been identified by the GA process, the corresponding best design variables are obtained by decoding. VI. NUMERICAL EXAMPLES To demonstrate the performance of the proposed ROSP solution procedure, three numerical examples are presented: a 45-story shear building, a simply supported beam, and a planar truss. A. 45-Story Shear Building A numerical example of a 45-story shear building is presented to evaluate the ROSP performance. The details of this structure are as follows: the mass of the steel on the two sides is neglected, and the stiffness is 2000 N/m. The stories are regarded as a rigid body, and the mass of each story is 30 kg. A 5% uncertainty level exists for both the stiffness and mass elements. A schematic diagram is shown in Fig. 4. Nine candidate sensors are selected using the ROSP method discussed above. Furthermore, the first nine modes are sampled as the target modes, which are shown in Fig. 5. The design variable must be coded because of the number of candidates. The parameters of the GA are set as follows: the population size is 100; the maximum number of generations is 500; the mutation function is Gaussian; the selection function is uniform and stochastic; and the crossover function is scattered. The value weighting factor is 0.8 in this numerical example. The convergence curve of the fitness function of the GA is illustrated in Fig. 6. When the fitness function has converged, the final layout of nodes 5, 9, 14, 19, 24, 29, 34, 39, and 45 is obtained by the proposed ROSP method. Fig. 5. First nine mode shapes of 45-story shear building. (a) 1st-3rd mode shapes. (b) 4th-6th mode shapes. (c) 7th-9th mode shapes. Compared to an uncertain case, the layout of nodes 4, 9, 14, 19, 25, 29, 34, 39, and 45 is achieved by the DOSP method for a deterministic case. These two final layouts using different OSP methods are listed in TABLE I. From TABLE I, it is clear that the final layouts are not the same. The locations obtained by ROSP are more robust. To verify the results of the ROSP method, four criteria are applied to evaluate the sampled locations obtained by DOSP and ROSP. These are as follows. Fisher determinant Qs As the fitness function in this paper is based on the Fisher determinant Qs, we consider this value as a criterion. Higher values of the Fisher determinant correspond to better sensor placement layouts.

7 YANG et al.: ROSP FOR UNCERTAIN STRUCTURES WITH INTERVAL PARAMETERS 2037 TABLE II FOUR OSP CRITERIA FOR 45-STORY SHEAR BUILDING Fig. 6. Convergence process of GA optimization in 45-story shear building. TABLE I RESULTS OF SENSOR PLACEMENT BY DOSP AND ROSP IN THREE NUMERICAL EXAMPLES Fig. 7. MAC values given by different methods in 45-story shear building. (a) DOSP. (b) ROSP. Ratio of singular values The ratio of the maximum and minimum singular values is selected as the second criterion for evaluating the sampled locations. This ratio is calculated as: β = λ max (28) λ min where λ min and λ max are the minimum and maximum singular values of FIM, respectively. Lower values of β correspond to better sensor placement layouts. Maximum off-diagonal MAC value The MAC index is used to evaluate the sampled modes. To reflect the inherent properties of the structural dynamics, the space angles of all sampled modes must be as large as possible [13]. Based on modal orthogonality, the MAC matrix reflects the relationship between two modes [25]. ( ) ϕ T 2 MAC ij = i ϕ j ( ) ( (29) ϕ T i ϕ i ϕ T j j) ϕ Lower values of the maximum off-diagonal MAC correspond to better sensor placement layouts. Mean off-diagonal MAC value Similar to the third criterion mentioned above, the mean offdiagonal MAC value can be easily computed and understood. Lower values of the mean off-diagonal MAC correspond to better sensor placement layouts. One group of random stiffness and mass parameters are applied in this example. As a sample of the associated uncertainties, this random selection of all possible parameters can be used to evaluate the effectiveness of the ROSP approach. The above four criteria are used to evaluate the sensor placement Fig. 8. Fitness value and interval for DOSP and ROSP in three numerical examples. obtained by the proposed ROSP method and for comparison the DOSP method. The results of the four OSP criteria for a 45-story shear building are listed in Table II, and a MAC graph is also presented in Fig. 7. All four criteria indicate that the ROSP layout is better than the DOSP layout. Therefore, we can draw the clear conclusion that if the nine sensors are placed according to the ROSP (nodes 5, 9, 14, 19, 24, 29, 34, 39, and 45) in this structure with uncertainty, then the performance and effectiveness of the sensor placement in the optimal ROSP solution is better than that of the DOSP solution and other positions. The above results illustrate the performance and effectiveness of the optimal solution using the ROSP method for SHM when uncertainties exist in the structure, particularly with respect to the verification using these four criteria. Furthermore, the fitness values and intervals for DOSP and ROSP are shown in Fig. 8, where the lower bound represents the fitness function as mentioned above. The central value and radius can be regarded as α f c ( d, b c) /f and (1 α) f (d) / f respectively. The optimal layout identified using the ROSP

8 2038 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, 2018 Fig. 9. Schematic diagram of simply supported beam. Fig. 11. beam. Convergence process of GA optimization for simply supported Fig. 10. First six mode shapes of simply supported beam. method can be compared with the DOSP layouts in Fig. 8. Although the central fitness value of ROSP is slightly lower than that of DOSP, the fluctuations in ROSP are much smaller. This means that the influence of uncertainty on ROSP is less obvious compared with DOSP. We can conclude that uncertainties may induce fluctuations in the fitness function, and that compromises between performance and robustness can be controlled by a weighting factor. B. Simply Supported Beam This section presents the example of a simply supported beam. The details of this structure are as follows: Young s modulus E = 210 GPa, section height h = 0.02 m, section width b = 0.02 m, length l = 1 m, and mass density ρ = 7670 kg/m 3. Both the Young s modulus and mass density have uncertainties at the 2% level. A schematic diagram is shown in Fig. 9. Six candidate sensors are selected using the ROSP method discussed above. Moreover, the first six modes are sampled as the target modes, and are presented in Fig. 10. The design variable should be coded according to the number of DOFs and sensors. The GA parameters are set as in the previous example. The value weighting factor is 0.8 in this numerical example. The convergence curve of the GA fitness function for ROSP is illustrated in Fig. 11. When the fitness function has converged, the final layout for nodes 6, 14, 21, 29, 36, and 44 is obtained by the proposed ROSP method. Compared to the uncertain case, the layout of nodes 6, 13, 21, 29, 37, and 44 is given by the DOSP method for the deterministic case. Moreover, these two final layouts are both listed in TABLE I using different OSP methods. The ROSP solution has better robustness than others. To verify the veracity and validity of the proposed method, four criteria are applied to evaluate the sampled locations obtained by DOSP and ROSP. As a sample of the uncertainties, a random selection from all possible parameters can be applied Fig. 12. MAC values given by different methods for simply supported beam. (a) DOSP. (b) ROSP. TABLE III THREE OSP CRITERIA FOR SIMPLY SUPPORTED BEAM to evaluate the effectiveness of the ROSP approach. The four criteria are used to evaluate the sensor placement obtained by the proposed ROSP method and, for comparison, the DOSP method. The four criteria results for the beam are listed in Table III. We can see that all four criteria indicate that the ROSP layout is better than the DOSP layout. Therefore, we can draw the clear conclusion that, if the sensors are placed according to the ROSP (namely, at nodes 6, 14, 21, 29, 36, and 44), the performance and effectiveness of the sensor placement will be better than that given by DOSP, which can be proved by the central fitness values and fluctuations shown in Fig. 8. C. Planar Truss This section presents the third numerical example of a planar truss. The details of this structure are as follows: Young s modulus E = 210 GPa, section area A = 1cm 2, and mass density ρ = 7670 kg/m 3. The length of each horizontal and upright bar element is l = 0.3 m. The Young s modulus and mass density have uncertainties at the 5% level.

9 YANG et al.: ROSP FOR UNCERTAIN STRUCTURES WITH INTERVAL PARAMETERS 2039 Fig. 13. Schematic diagram of planar truss. Fig. 15. Convergence process of GA optimization in planar truss. Fig. 16. MAC values given by different methods for planar truss. (a) DOSP. (b) ROSP. Fig. 14. First eight mode shapes of planar truss. TABLE IV FOUR OSP CRITERIA IN PLANAR TRUSS A schematic diagram is shown in Fig. 13. The truss consists of 151 bar elements and 121 free DOFs. The first eight modes are sampled as the target mode, as presented in Fig. 14. Furthermore, eight candidate sensors are selected using the ROSP method discussed above. The design variable should be coded according to the number of DOFs and sensors. The value weighting factor in this numerical example is again 0.8. The convergence curve for the GA fitness function in GA for ROSP is illustrated in Fig. 15. When the fitness function has converged, the final layout of DOFs 16, 34, 41, 54, 68, 90, 104, and 119 is obtained by the proposed ROSP method. Compared to the uncertain case, the layout of DOFs 16, 34, 47, 52, 68, 90, 104, and 121 is given by the DOSP method in the deterministic case. These two final layouts are listedintableiandshowninfig.13(greenandorange arrows denote sampling DOFs obtained by DOSP and ROSP, respectively). Four of the sensor positions given by ROSP are different to those of the DOSP results. To verify the veracity and validity of the proposed method, the above four criteria are used to evaluate the sensor placement obtained by the proposed ROSP method and the DOSP method with a random selection of all possible parameters. The results for OSP criteria in the beam are given in Table IV, and a MAC graph is presented in Fig. 16. According to all four criteria, the ROSP layout is better than the DOSP layout. The optimal layout identified using the ROSP method can be compared with the DOSP layouts in Fig. 8. With much smaller fluctuations and a slightly lower central fitness value, the ROSP solution has the better anti-uncertainty performance. Therefore, we can draw the clear conclusion that if the eight sensors are placed according to the ROSP, the performance and effectiveness of sensor placement for the optimal ROSP solution is better than in the DOSP and other positions. VII. CONCLUSION This study proposes an ROSP method for SHM that accounts for parameter uncertainties. Because of the scarcity of

10 2040 IEEE SENSORS JOURNAL, VOL. 18, NO. 5, MARCH 1, 2018 accurate statistical distributions in models and responses, it is difficult to quantify the uncertainty of structural information in engineering. Therefore, the proposed ROSP method considers the uncertainties as interval numbers. Based on an interval analysis approach, the FIM was extended to the uncertain case, and the bounds of the IFIM eigenvalues were derived. The fitness function is defined as the IFIM determinant, which consists of a central interval and radius value. By applying normalization and using weighted coefficients, the ROSP problem can be transformed into a deterministic optimization process. To obtain the best ROSP layout (that is, the optimized solution), a GA was adopted in the optimization process. The propagation of uncertainties was investigated for the fitness function and ROSP using GA. 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11 YANG et al.: ROSP FOR UNCERTAIN STRUCTURES WITH INTERVAL PARAMETERS 2041 [36] A. J. Lee and U. M. Diwekar, Optimal sensor placement in integrated gasification combined cycle power systems, Appl. Energy, vol. 99, no. 6, pp , Nov [37] L. Vincenzi and L. Simonini, Influence of model errors in optimal sensor placement, J. Sound. Vibrat., vol. 389, no. 17, pp , Feb [38] A. Molyboha and M. Zabarankin, Stochastic optimization of sensor placement for diver detection, Oper. Res., vol. 60, no. 2, pp , [39] R. F. Guratzsch and S. Mahadevan, Structural health monitoring sensor placement optimization under uncertainty, AIAA J., vol. 48, no. 7, pp , [40] R. D. Carr et al., Robust optimization of contaminant sensor placement for community water systems, Math. Programm., vol. 107, nos. 1 2, pp , [41] J. Blesa, F. Nejjari, and R. Sarrate, Robustness analysis of sensor placement for leak detection and location under uncertain operating conditions, Proc. Eng., vol. 89, no. 2, pp , [42] B. Li and A. D. Kiureghian, Robust optimal sensor placement for operational modal analysis based on maximum expected utility, Mech. Syst. Signal Process., vol. 75, pp , Jun [43] R. Castro-Triguero, S. Murugan, M. I. Friswell, and R. Gallego, Robustness of optimal sensor placement under parametric uncertainty, Mech. Syst. Signal Process., vol. 41, nos. 1 2, pp , [44] I. Elishakoff, Three versions of the finite element method based on concepts of either stochasticity, fuzziness, or anti-optimization, Appl. Mech. Rev., vol. 51, no. 3, pp , [45] Z. Qiu and I. Elishakoff, Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis, Comput. Method Appl. Mech. Eng., vol. 152, nos. 3 4, pp , [46] C. Yang and Z. X. Lu, An interval effective independence method for optimal sensor placement based on non-probabilistic approach, Sci. China Technol. Sci., vol. 60, no. 2, pp , [47] Y. Li, X. Wang, R. Huang, and Z. Qiu, Actuator placement robust optimization for vibration control system with interval parameters, Aerosp. Sci. Technol., vol. 45, no. 8, pp , Sep [48] J. S. Sim, Z. P. Qiu, and X. J. Wang, Modal analysis of structures with uncertain-but-bounded parameters via interval analysis, J. Sound. Vibrat., vol. 303, nos. 1 2, pp , [49] Z. P. Qiu, Convex Method Based on Non-Probabilistic Set-Theory and its Application. Beijing, China: National Defence Industry Press, [50] T.-H. Yi, H.-N. Li, and M. Gu, Optimal sensor placement for health monitoring of high-rise structure based on genetic algorithm, Math. Problem Eng., vol. 2011, Mar. 2011, Art. no Chen Yang received the bachelor s degree in engineering mechanics and the master s degrees in solid mechanics from Beihang University, Beijing, China, in 2010 and 2013, respectively, where he is currently pursuing the Ph.D. degree in solid mechanics. He published about ten research papers in the journals particularly in the area of structural health monitoring and its applications. His research covers structural health monitoring, optimal sensor placement, structural dynamics, and thermal control. Zixing Lu received the bachelor s and Ph.D. degrees in science from the Mechanics Department, Peking University, in 1982 and 1995, respectively, and the Post-Doctoral degree from Beihang University (BUAA) in 1997, where he received the Professorship in He is currently a Professor and Doctoral Supervisor with BUAA. His major research fields include the design and optimization of stiffness and strength for novel composite materials, the damage analysis and reliability assessment for composite materials, and the mechanical characterization for micro-nano materials. Up to now, he has published over 200 papers, and over 130 papers have been indexed by SCI, EI, and ISTP. Zhenyu Yang received the bachelor s and master s degrees from Beihang University (BUAA), in 2001 and 2004, respectively, the Ph.D. degree from the Institute of Mechanics, Chinese Academy of Sciences, in 2008, and the Post-Doctoral degree from BUAA, in He is currently an Associate Professor with BUAA. His major research fields include computational solid mechanics, composite material mechanics, and micro-nano mechanics. He published over 40 research papers in the reputed journals, such as Carbon, Scientific Reports, IJSS, the Journal of Applied Physics, Materials Science and Engineering: A, Composites Part B: Engineering, andcomputational Material Science.

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