Optimal placement of sensors for structural health monitoring using

INSTITUTE OF PHYSICS PUBLISHING

SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 13 (2004) 528–534

PII: S0964-1726(04)76139-X

Optimal placement of sensors for structural health monitoring using improved genetic algorithms H Y Guo, L Zhang, L L Zhang and J X Zhou School of Architecture Engineering and Mechanics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China E-mail: [email protected]

Received 28 March 2003, in final form 5 January 2004 Published 19 April 2004 Online at stacks.iop.org/SMS/13/528 DOI: 10.1088/0964-1726/13/3/011

Abstract The global optimization of sensor locations for structural health monitoring systems is studied in this paper. First, the performance function based on damage detection is presented. Then, genetic algorithms (GAs) are adopted to search for the optimal locations of sensors. However, the simple GAs can result in infeasible solutions to the problem. Some improved strategies are presented in this paper, such as crossover based on identification code, mutation based on two gene bits, and improved convergence. The analytical results from the improved genetic algorithm are compared with the penalty function method and the forced mutation method. It is concluded that the convergence speed with the proposed improved genetic algorithm is faster than that with the penalty function method and the forced mutation method, and the result of placement optimization is better.

1. Introduction All load-carrying structures, such as aircraft, spacecraft, bridges, and offshore platforms, continuously accumulate damage during their service life. Any crack or local damage in a structure may affect the structural safety. So a structural monitoring system is needed. The location optimization of sensors is a crucial problem in a structural monitoring system. Taking the cost of sensors into account, it is uneconomical to install sensors on every part of a structure. For a limited number of sensors, two questions naturally arise: which type of sensor placement optimization performance index should be adopted, and which algorithm can be used? A placement optimization performance index based on damage detection and an improved genetic algorithm are presented to answer these questions. Methods have been developed to place sensors in an optimal fashion to address the identification and control of dynamic structures by Udwadia and Garba [1] and Lim [2]. Kammer [3] proposed an effective independence algorithm based on the contribution of each sensor location to the linear independence of the identified modes. The initial 0964-1726/04/030528+07$30.00 © 2004 IOP Publishing Ltd

candidate set of sensor locations was quickly reduced to the number of available sensors. Hemez and Farhat [4] extended the effective independence method in an algorithm where sensor placement was achieved in terms of the strain energy contribution of the structure. Miller [5] computed a Gaussian quadrature formula using the functional gain as a weight function, and thought that the nodes of the quadrature formula gave the optimal locations for sensors. Hiramoto et al [6] used the explicit solution of the algebraic Riccati equation to determine the optimal sensor/actuator placement for active vibration control. Wouwer et al [7] presented an optimality criterion for the selection of optimal sensor locations; the criterion was based on a measure of independence of the sensor responses. Worden and Burrows [8] used a number of different methods to determine an optimal sensor distribution based on the curvature data. Tongpadungrod et al [9] used eigenvalues derived from principal component analysis (PCA) as the performance evaluators, and described an optimization technique through sensor placement using a search algorithm based on a genetic algorithm. However, these placement optimization criteria focused on maximizing either the controllability or observability of the system, not on

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Optimal placement of sensors for structural health monitoring using improved genetic algorithms

damage detection. Cobb and Liebst [10] and Shi et al [11] have reported the optimal sensor placement for the purpose of detecting structural damage. No other work is known to the authors emphasizing optimum sensor locations based on their ability to localize damage from test modal data. To place the sensors simultaneously, a genetic algorithm is an effective method. GAs are global probabilistic search algorithms inspired by Darwin’s survival-of-the-fittest theory. A distinguishing characteristic of GAs is that the algorithms work with coding of the parameter set, not the parameters themselves. Generally, the binary code is used. Because of the discrete nature of coding, the algorithms are a perfect choice for those problems with discrete variables. GAs have been widely used in sensor placement type problems [9, 12]. They have been used to search for the optimal locations of actuators in active vibration control [13, 14]. Another type of problem where GAs have been successfully utilized is the placement of vibration isolators to reduce the transmissibility of undesirable vibrations to an optical laboratory table [15]. However, these methods often produce some invalid strings in the evolution process. In this paper, a sensor placement optimization performance index based on damage detection is presented, and GAs are employed to determine the location of sensors on the structures. It is considered that the number of sensors needed is often fixed. The simple GAs cannot avoid producing some invalid strings that violate the constraints caused by the fixed number of sensors in the evolution process [12]. Constraints are mostly handled by using penalty functions, which penalize infeasible solutions by reducing their fitness values in proportion to their degrees of constraint violation. However, the performance of the penalty function depends on the selection of penalty parameters [16, 17]. Liu and Zhang [18] presented the forced mutation method, which forcibly converts the generated invalid strings into valid strings to meet the constraints. In nature, the forced mutation method greatly increases the mutation probability. Therefore, the forced mutation method is close to random search methods. In order to avoid these problems, an improved genetic algorithm (IGA) is presented in this paper. It will be demonstrated that the IGA is a more effective optimization technique than the aforementioned methods.

2. Theory In the theoretical development that follows, structural damage refers to changes in the stiffness properties of the structure with no change in the mass property. The stiffness changes are small and would not cause a change in the connectivity of the structure. The proposed method would be applicable to a lightly damped structure where stiffness changes would not significantly affect the damping property of the structure. For a small perturbation in stiffness in such an n-degree-of-freedom dynamic system where structural damping is ignored, [(K + K ) − (λi + λi ) M] (φi + φi ) = 0

(1)

in which K and M are the stiffness matrix and the mass matrix of the structure; λi and φi are the i th eigenvalue and eigenvector; K , λi and φi are small changes in the stiffness matrix and the i th frequency and mode shape,

respectively, due to the damage. Neglecting second-order terms, equation (1) becomes (K − λi M) φi = λi Mφi − K φi .

(2)

We express φi as a linear combination of mode shapes of the original system based on the method presented by Fox and Kapoor [19]: n φi = dik φk (3) k=1

where dik is a scalar factor and n is the total number of modes of the original n-DOF system. Substituting equation (3) into (2) and premultiplying both sides of equation (2) by φrT (r = i ), we have n

dik φrT (K − λi M) φk = λi φrT Mφi − φrT K φi ;

(4)

k=1

with the orthogonal relationship, equation (4) can be simplified as φ T K φi dir = − r . (5) λr − λi For the case of r = i , and with no change in mass matrix of the system, it can be found that drr = 0 using the orthogonal relationship φrT Mφi = 1. Therefore, equation (3) can be written as n −φrT K φi φr . (6) φi = λr − λi r=1 r=i

It is difficult to model the damage in sufficient detail for a general type of damage. Here, it is assumed that the reduction of structural stiffness due to damage as the summation of each elemental stiffness matrix multiplied by a damage coefficient, K , can be expressed as K =

L

αk K k ,

(−1 αk 0)

(7)

k=1

in which K k and αk are the kth elemental stiffness matrix and its damage coefficient, respectively. This model is suitable for most types of damage in an actual structure. It works even for damage whose change is not proportional to the elemental stiffness, because this assumption gives only a small error for a large structure and it does not change the essence of the requirements in the damage localization. Substituting equation (7) into (6), the change of the i th mode shape can be represented as a summation of the contribution of each damage to the mode shape in the structure. Thus equation (6) becomes φi =

L

αk

k=1

F(K ) =

n r=1 r=i

−φrT K 1 φi λr − λi

n −φrT K k φi φr = F(K )δ A λr − λi r=1

(8)

r=i

φr

δ A = { α1

n −φrT K 2 φi φr λr − λi r=1

···

r=i

α2

···

α L }T

n −φrT K L φi φr λr − λi r=1

r=i

(9) (10)

where F(K ) is the matrix of sensitivity coefficients of the i th mode shape changes with respect to the damage vector δ A. If several modes are used, φi would become the measured mode shape change matrix and F(K ) the sensitivity matrix for the selected modes. 529

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3. Sensor placement optimization performance function In order to acquire sufficient damage information, it is expected that the mode shape change can reach the same maximum value for the same damage coefficients. Here, premultiplying both sides of equation (8) by φi , we have (11) φiT φi = δ AT F(K )T F(K ) δ A. In equation (11), the Fisher information matrix A0 can be defined as follows: A0 = F(K ) F(K ). T

(12)

The Fisher information matrix is a summation of the contribution of each degree of freedom or sensor location to the mode shapes of the structure. Maximizing the Fisher information matrix A0 will lead to the best estimate of damage coefficients [1]. Shi et al [11] utilized the following matrix E to place sensors, and the contribution from each degree of freedom is referred to as the diagonal element of the matrix: −1 F(K )T . E = F(K ) F(K )T F (K )

L L

F(K )si F(K )s j

t1 + t2 + · · · + tn = q.

(15)

Thus, the sensor placement optimization problem using GAs can be expressed as maximizing equation (14) f =

L L

F(K )si F(K )s j

i=1 j =1 s∈m

under constraint (15) t1 + t2 + · · · + tn = q.

(13)

However, it is difficult to guarantee that the matrix [F(K )T F(K )] is always nonsingular. The inverse of the matrix [F(K )T F(K )] often is difficult to solve. Here, from equation (12), we have a performance function or objective function based on damage detection, as follows: f =

freedom to place sensors, the coding length of a string is n. If the value of the i th bit position of a string is 1, this denotes that a sensor is located on the i th degree of freedom; in contrast, if the value of the i th bit position is 0, this denotes no sensor on the i th degree of freedom. For example, 001001100100 denotes that sensors are located on the third, sixth, seventh, and tenth degrees of freedom. If the number of sensors needed is q and ti denotes the gene value of the i th bit position of a string, the constraint can be expressed as

(14)

i=1 j =1 s∈m

where s ∈ m denotes that s is confined to all locations where sensors are placed. Maximizing the objective function value will lead to the best estimate of damage coefficients.

Initial population GAs start from an initial population of individuals as initial variable. Each individual in the initial population should meet the constraint. Thus, it is easy for GAs to reach the global optimum solution. Selection The selection procedure used by GAs is based on the fitness of each individual. There are a number of reproduction schemes commonly used in GAs. These include proportionate reproduction, ranking selection, tournament selection, steadystate selection, and greedy over-selection. Tournament selection can be used.

4. Genetic algorithms

Crossover

In this optimization method, information about a problem, such as variable parameters, is coded into a genetic string known as a chromosome (individual). Each of these chromosomes has an associated fitness value, which is usually determined by the objective function to be maximized or minimized. Each chromosome contains substrings known as genes, which contribute in different ways to the fitness of the chromosome. The genetic algorithm proceeds by taking a population, which is comprised of different chromosomes with highest fitness, and increasing the average fitness of each successive generation. Only the fittest chromosomes pass to successive generations. Now we introduce some basic concepts and operators of GAs before the description of the suggested algorithm.

Crossover is the operator that produces new individuals (offspring) by exchanging some bits of a couple of randomly selected individuals (parents). The general form of the operator is one-point crossover, described as follows. First, a cutting position is chosen at random between the first and the last bit of the parents, and the parents are thus divided into two parts by the cutting. Then, the two offspring are generated by each taking the first part from one parent and the second from the other. For example, the constraint is q = 4; assume a number 4 is chosen at random as the cutting position. The one-point crossover operation is shown in figure 1. We can find that even if the parent strings satisfy the constraint, the generated offspring still violate the constraint. Therefore, we will improve the crossover operator in the following section.

Coding

Mutation

An essential characteristic of a GA is the coding of the variables that describe the problem. A binary coding method can be used. The method is to transform the variables to a binary string of specific length. If there are n optional degrees of

Mutation operates on a single individual with a small probability. With this operation, one or more bits are chosen at random from the individual and changed into a different symbol. This is shown in figure 2.

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Optimal placement of sensors for structural health monitoring using improved genetic algorithms

Parent1

0 1 0 0 1 1 0 0 1 0

Parent1

0 1 0 0 1 1 0 0 1 0

Parent2

1 0 1 0 0 0 1 0 1 0

Parent2

1 0 1 0 0 0 1 0 1 0

Identification code

0 0 1 0 1 0 0 0 0 0

Offspring1

0 1 0 0 0 0 1 0 1 0

Offspring1

0 1 1 0 0 1 0 0 1 0

Offspring2

1 0 1 0 1 1 0 0 1 0

Offspring2

1 0 0 0 1 0 1 0 1 0

Figure 1. Sketch of one-point crossover.

Figure 3. Sketch of improved crossover.

Parent3

0 1 0 0 1 1 0 0 1 0

Parent3

0 1 0 0 1 1 0 0 1 0

Offspring3

0 1 0 0 0 1 0 0 1 0

Offspring3

0 1 1 0 0 1 0 0 1 0

Figure 2. Sketch of basic mutation.

Similarly, in order to meet the constraint, the mutation operation should be improved, too. In order to apply a genetic algorithm to the sensor optimization placement problem, we have the following steps. (1) Create an initial population randomly, and calculate the fitness values of the strings. (2) Select the fittest individuals according to fitness values, and apply the crossover operation and mutation operation. (3) Calculate the fitness values of the new strings. (4) Repeat (2) and (3) until the termination criterion is met.

5. Improved genetic algorithm The sensor placement optimization problem is a constraint optimization problem. It is difficult for simple GAs to solve the problem. Although penalty function methods can handle the constraints, they cannot avoid the generation of individuals that violate constraints. Moreover, the performance of the penalty function depends on the selection of penalty parameters [16, 17]. For this problem, the appropriate penalty function can be expressed as f +C p=q f1 = (16) 2 p = q f + C − α ( p − m) where f is the fitness function; f 1 is the penalized fitness; C is a positive number, that guarantees the penalty function value >0; α is a penalty parameter; p is the actual number of sensors; q is the number of sensors needed. If the penalty is too large, the search process may converge too quickly, which does not allow the GA to exploit various combinations of strings. If the penalty is too small, the convergence process may be too slow and the computational costs could be high. So it is difficult to select an appropriate penalty parameter. Liu and Zhang [18] presented the forced mutation method to avoid the generation of individuals that violate the constraints. However, it greatly increases the mutation probability, and makes GAs tend to random search methods. In order to avoid these problems, some improved strategies are presented as follows.

Figure 4. Sketch of two-gene-bit mutation.

Improved crossover Consider the constraint equation (15). The general crossover operator will produce some offspring that violate the constraint. This paper utilizes the characteristic that the better characteristics of parents may be inherited by offspring in nature and assumes that the constraint equation (15) is the better characteristic shared by parents and offspring. The improved method is that a binary string which only includes two coding bit positions whose values are 1 is randomly produced, such as 0010100000. The string length is equal to the length of the coding individuals, and the bit positions whose coding value is 1 are located randomly in the string. It can be assumed that the two bit positions whose coding value is 1 are the i th and j th bit positions respectively, and two individual (parents) are Parent1 (P1) and Parent2 (P2). If the condition is satisfied t(P1)i + t(P1)i+1 + · · · t(P1) j = t(P2)i + t(P2)i+1 + · · · t(P2) j (17) i.e. the summation of the two individual gene values from i th to j th bit positions are equal, the binary string is named the identification code. If the equation (17) is not met, a new binary string, which includes two coding bit positions whose values are 1, is randomly created again until equation (17) is met. In the identification code, the bit positions whose values are 1 are named identification positions. The segments between identification positions of parent strings are swapped to create two new strings. The crossover scheme is illustrated in figure 3. Two-gene-bit mutation First, the mutation operation randomly selects one gene position from the ones whose value is 1 in a string. Then the mutation operation randomly selects another gene position from the ones whose value is 0 in the string. Finally, we swap the positions of the two genes to produce an offspring. The mutation scheme is shown in figure 4. Through the operation of above-mentioned crossover and mutation, if the parents satisfy the constraint, the offspring should satisfy the constraint too. 531

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Table 1. Calculational results of ten runs by the improved genetic algorithm, the penalty function method and the forced mutation method. GA (forced mutation)

IGA

No (1)

Number of convergence generations (2)

Fitness value (×10−9 ) (3)





1 2 3 4 5 6 7 8 9 10 Average

119 101 200 200 68 200 92 200 110 200 149

1.6640 1.6712 1.6740 1.6775 1.6677 1.6820 1.6586 1.6623 1.6820 1.6931 1.6732

37 200 121 66 42 165 200 96 121 165 121.3

1.6931 1.6931 1.6931 1.6931 1.6931 1.6931 1.68 1.6931 1.6931 1.6931 1.6918

44 56 61 43 44 56 63 61 59 84 57.1

1.6931 1.6931 1.6931 1.682 1.6931 1.6931 1.6931 1.6931 1.6931 1.6931 1.6920

2

2

4

3

7 8

4 3

8

5

10

7

12

23

19 15

22

26 24

20

9

13

27 28

21

16 14

10

17 18

11 9

5

12 13

6

1

1

6

31

1m

GA (penalty function)

29 25

11

30

14

1m

Figure 5. Two-dimensional truss structure.

Convergence criterion The two termination criteria are given as follows. (1) Consider the two variables maximum fitness fitmax and average fitness fitavg . A convergent criterion is given as fitmax − fitavg ε (18) where ε is a small number. (2) In order to avoid redundant iteration, an appropriate number w is selected. If the fittest individual in the population does not change in continuous w iterations, we think that the optimization value is found, and the genetic process can be stopped automatically. The number w should be large enough to avoid the premature convergence. If either of the two convergence criteria is met, the genetic operation should be terminated.

6. Examples and analysis In this section, the optimization of sensor locations is studied using a two-dimensional truss structure shown in figure 5. The finite-element model of the truss consists of 31 classical consistent elements, 14 nodes, and 28 degrees of freedom. The following properties of the structure and parameters of GAs were used in the analysis: elastic modulus E = 70 GPa, material density ρ = 2800 kg m−3 , length of rod element l = 1 m; population size pop = 40, probability of crossover pc = 0.9, probability of mutation pm = 0.1, convergence parameters w = 30 and maximum number of generations 200. Here, 15 sensors need to be located on the structure. 532

The first three modes are used to determine sensor locations. In order to compare with other methods that can be applied to search for the optimal sensor locations, such as the penalty function method and the forced mutation method [18], we ran each method ten times with a different initial population, which was randomly created. The penalty function of equation (16) is applied, and a good penalty parameter is selected as α = 3×10−10 . The penalty parameter value approximates to the maximum change of fitness value caused by removing a sensor. So the penalty parameter can have a good penalty effect, and it makes the penalty function value of invalid individuals that avoid the constraint far less than that of the valid individuals that satisfy the constraint. Thus, the invalid individuals will be eliminated. The simulation results are listed in table 1. It is observed that the number of convergence generations using IGA is less than those both using the penalty function method and the forced mutation method. The average number of convergence generations using IGA is 57.1, but the average numbers of convergence generations using the penalty function method and forced mutation are 149 and 121.3 respectively. The convergence speed of our improved GA is far higher than that of the penalty function method and the forced mutation method, with about 61.7% and 52.9% savings in computational iterations to reach a satisfactory solution respectively. It is difficult for the penalty function method to reach a satisfactory solution. Although the forced mutation method can reach a satisfactory solution, the convergent speed is slower than that of IGA. From table 1, the insensitivity of the numbers of convergence generations to the different initial populations is very apparent for IGA. In contrast, the forced mutation method and penalty function method show very

Optimal placement of sensors for structural health monitoring using improved genetic algorithms maximum fitness average fitness

maximum fitness average fitness

1.8 1.6

1.4

1.4

Fitness ( x 10-9 )

1.6

Fitness ( x 10-9 )

1.2 1.0 0.8 0.6

1.2 1.0 0.8 0.6

0.4

0.4

0.2

0.2 0.0

0.0 0

10

20

30

40

0

50

20

40

60

Generation

Figure 6. A typical fitness curve of improved genetic algorithm. maximum fitness average fitness

1.4

Fitness ( x 10-9 )

1.2 1.0 0.8 0.6 0.4 0.2 0.0 20

30

40

120

Table 2. Optimal placement results for five, ten, and 15 sensors, respectively. 15 sensors

10

100

Figure 8. A typical fitness curve of the penalty function method.

1.6

0

80

Generation

50

60

70

Generation

Figure 7. A typical fitness curve of the forced mutation method.

high variation when the initial population is varied. This confirms the higher robustness of the proposed method over other methods. The typical fitness curves of the IGA, forced mutation method and penalty function method are depicted in figures 6– 8, respectively. From figure 6, we can see that the maximum fitness value using IGA quickly tends to a constant and the average fitness value steadily tends to maximum fitness along with increasing number of generations. It shows a good characteristic of convergence. From figure 7, it can be observed that in the forced mutation method the fluctuation of the average fitness curve is sharper than that in IGA. This may be because the higher actual mutation probability caused by the forced mutation of plenty of invalid individuals makes the method tend to a random search method. From figure 8, we can observe that in the penalty function method at the beginning the average fitness value has a sudden reduction, then the average fitness unsteadily tends to maximum fitness with increasing number of generations. The reason is that though the initial population meets the constraint caused by the fixed number of sensors, plenty of invalid individuals

10 sensors

5 sensors

Number (1)

Node (2)

DOF (3)

Node (4)

DOF (5)

Node (6)

DOF (7)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3 4 4 5 5 6 7 7 8 9 10 11 12 14

3 6 7 8 9 10 12 13 14 16 18 20 22 24 27

3 4 5 6 7 8 9 10 11 12

6 8 10 12 14 16 18 20 22 24

5 6 9 10 11

10 12 18 20 22

that violate the constraint are produced after simple crossover and mutation operations. Due to the effect of the penalty function, the fitness value of the invalid individuals is greatly reduced. So at the beginning the average fitness suddenly falls. Moreover, the invalid individuals that violate the constraint not only hold the position of valid individuals, but also waste plenty of computational time. The IGA does not only avoid the generation of invalid individuals, but also avoids the random search tendency. Therefore, our proposed algorithm is quite superior to the others in effectiveness and efficiency. The optimal placement of five, ten, and 15 sensors is solved by using the improved genetic algorithm, respectively. The results are listed in table 2. From table 2, it is observed that for the different numbers of sensors the locations of fewer sensors are included in the locations of more of sensors. This shows that the optimization results of different numbers of sensors have an inheritable characteristic, and demonstrates that the convergent result is credible. 533

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7. Conclusions This paper presents a sensor placement optimization performance index based on damage detection and some improved strategies in GAs for sensor location optimization. The simple genetic operation generally generates many invalid individuals that avoid constraints when the number of sensors is fixed. So it is difficult for the general GA to solve the constrained optimization problem. Although the penalty function method can be applied to the problem, it cannot avoid the generation of the invalid individuals. Thus it will not lead to a good result or high efficiency. The forced mutation method avoids the generation of invalid individuals, but the efficiency is not high due to the higher actual mutation probability. The IGA avoids the above-mentioned problems. Therefore, the convergence result and efficiency of IGA is better than those both of the penalty function method and the forced mutation method. The research work shows the validity and effectiveness of IGA for sensor placement required in a structural health monitoring system. The numerical example and analysis demonstrate that IGA is able to determine the optimal sensor locations based on damage detection. Moreover, the method can be applied to similar constrained optimization problems.

Acknowledgments This work has been supported by the Natural Science Foundation of Shaanxi Province, People’s Republic of China (grant No 2002E2 06).

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