Title of Paper s1

Optimal experimental design for structural health monitoring applications

H.F. Lam City University of Hong Kong, Department of Building and Construction, Hong Kong, China C. Papadimitriou, E. Ntotsios University of Thessaly, Department of Mechanical & Industrial Engineering, Volos 38334, Greece

ABSTRACT: Successful structural health monitoring and condition assessment depends to a large extent on the sensor and actuator networks place on the structure as well as the excitation characteristics. An optimal experimental design methodology deals with the issue of optimizing the sensor and actuator network, as well as the excitation characteristics, such that the resulting measured data are most informative for monitoring the condition of the structure. Theoretical and computational issues arising in optimal experimental design are addressed. The problem is formulated as a multi-objective optimization problem of finding the Pareto optimal sensor configurations that simultaneously minimize appropriately defined information entropy indices related to monitoring multiple probable damage scenarios. Asymptotic estimates for the information entropy, valid for large number of measured data, are used to rigorously justify that the selection of the optimal experimental design can be based solely on nominal structural models associated with the probable damage scenarios, ignoring the details of the measured data that are not available in the experimental design stage. Heuristic algorithms are proposed for constructing effective, in terms of accuracy and computational efficiency, sensor configurations. The effectiveness of the proposed method is illustrated by designing the optimal sensor configurations for monitoring damage on a shear model of a building structure.

Keywords: Structural identification, experimental design, information entropy, sensor placement, Pareto op- tima. 1 INTRODUCTION The optimal sensor placement strategies depend on the class of mathematical models selected to rep- Information theory based approaches (e.g. Shah & resent structural behavior as well as the model pa- Udwadia 1978, Kammer 1991, Kirkegaard & rameterization within the model class. However, it is Brincker 1994) have been developed to provide ra- often desirable to use the measured data for selecting tional solutions to several issues encountered in the the most appropriate model class from a family of problem of selecting the optimal sensor configura- alternative model classes chosen by the analyst to tion. The optimal sensor configuration is taken as the represent structural behavior. Such classes may be one that maximizes some norm (determinant or linear (modal models or finite element models), non- trace) of the Fisher information matrix (FIM). linear elastic or inelastic, each one involving differ- Papadimitriou et al. (2000) introduced the informa- ent number of parameters. Model class selection is tion entropy norm as the measure that best corre- also important for damage detection purposes for sponds to the objective of structural testing, which is which the location and severity of damage are iden- to minimize the uncertainty in the model parameter tified using a family of model classes with each estimates. An important advantage of the informa- model class monitoring a specific region in a struc- tion entropy measure is that it allows one to make ture (Papadimitriou & Katafygiotis 2004) or incor- comparisons between sensor configurations involv- porating different mechanisms of damage. ing a different number of sensors in each configura- The objective in this work is to optimise the tion. Furthermore, it has been used to design the op- number and location of sensors in the structure such timal characteristics of the excitation (e.g. amplitude that the resulting measured data are most informat- and frequency content) useful in the identification of ive for estimating the parameters of a family of linear and strongly nonlinear models (Metallidis et mathematical model classes used for structural iden- al. 2003). tification and damage detection purposes. The problem of optimally placing the sensors in the structure sponse characteristics predicted by a particular mod- is presented for two cases: (i) identification of struc- el in the model class Μ , and r(s ) is a function of tural model (e.g. finite element) parameters or modal the prediction error parameters s , pq (q ) and model parameters (modal frequencies and modal ps (s ) are the prior distribution for the parameter damping ratios) based on acceleration time histories, sets q and s , respectively, N= N0 , N 0 is the and (ii) identification of structural model parameters number of response characteristics, N D is the num- based on modal data. Asymptotic estimates for the ber of measured data sets, and c is a normalizing information entropy, valid for large number of mea- constant chosen such that the PDF in integrates to sured data, are used to rigorously justify that the se- one. lection of the optimal experimental design can be For the case for which the response characteris- based solely on the nominal structural model from a tics consist of the response time histories data N 0 class, ignoring the details of the measured data that D={ xj ( kD t ) � R , j = 1,L , N0 , k 1, L , N D } are unavailable in the experimental design stage. A at N 0 measured DOFs, where N D is the number of heuristic algorithm is used for constructing effective the sampled data using a sampling rate Dt , then Pareto optimal sensor configurations that are superi- or, in terms of accuracy and computational efficien- N0 ND 1 1 1 轾 2 cy, to the Pareto optimal sensor configurations ap- J(q ; s )=邋 Jj ( q ), J j ( q ) = xˆ j ( k ) - x j ( k ; q ) N2 N 臌犏 proximated by genetic algorithms (Zitzler and Thiele 0 j=1s j D k = 1

1999), suitable for solving the resulting multi-objec- N ( ) = 0 N D x( k ; ) j= 1,L , N tive optimization problems. Damage detection re- rs j =1 s j , and j q , d , are sults on a shear model of a building are used to illus- the predictions of the sampled response time histo- trate the theoretical developments. ries obtained from a particular model in the model class Μ corresponding to a specific value of the parameter set q . 2 STRUCTURAL IDENTIFICATION AND For the case where the response characteristics DAMAGE DETECTION METHODOLOGY consist of modal data (k ) ( k ) N 0 D={w�r ,f r� R 1 r = ,,L , m 1 k ,, L } N D , (k ) (k ) Consider a parameterized class Μ of structural where wr are the modal frequencies and f r are models (e.g. a class of finite element models or a modeshape components at N 0 measured DOFs, m class of modal models) chosen to describe the input- is the number of observed modes and N D is the N output behavior of a structure. Let q R q be the number of modal data sets available, then vector of free parameters (physical or modal param- 2 eters) in the model class. A Bayesian statistical sys- (k ) ( k ) m 轾 N(k ) 2 N D圿 D rL0 r( ) - r tem identification methodology (Beck & Katafygio- 1犏 1 [wr (q ) - w r N 0 b f q f J ( ; ) =邋犏 + 2 q s 2 (k ) 2 2 (k ) tis 1998, Katafygiotis et al. 1998) is used to estimate ND N r=1犏s k = 1[ w圿 r s k = 1 犏wrf r f r the values of the parameter set q and their associat- 臌 ed uncertainties using the information provided from dynamic test data. For this, the uncertainties in the ( ) ( ) N d = where wr q and fr q R , r1,L , m , are values of the structural model parameters q are the predictions of the modal frequencies and mode- quantified by probability density functions (PDF) shapes obtained for a particular value of the model that are updated using the dynamic test data. The up- = + parameter set q , N m( N 0 1) is the number of dated PDF is then used for designing the optimal measured data per modal set, and sensor configuration. (k ) ( k ) T T br= f r(L0 f r )/ ( L 0 f r ) ( L 0 f r ) is a normalization According to the Bayesian structural identifica- constant that accounts for the different scaling be- tion methodology, assuming independent and zero- tween the measured and the predicted modeshape. e( k ) mean Gaussian prediction errors j with variance Damage detection is accomplished by introduc- 2 p( , | D ) s j , the updating PDF q s of the parameter ing a family of m model classes Μ⋯ Μ and asso- = ( ,L , ) 1 m sets q and s s1 sNo , given the measured ciating each model class to a damage pattern in the data D and the class of models Μ , takes the form structure, indicative of the location of damage. Each (Papadimitriou and Christodoulou 2007): model class Μ i is assumed to be parameterized by a number of structural model parameters qi scaling the stiffness contributions of a “possibly damaged”  禳 c 镲 ND N substructure, while all other substructures are as- p(q , s | D )= exp睚 - J ( q ; s ) p ( q ) p ( s ) ND N0 q sumeds to have fixed stiffness contributions equal to 2 ( ) 铪镲 2 ( p) r s those corresponding to the undamaged structure. Us- ing a Bayesian model selection framework, the prob- where J (q ; s ) is a measure of fit of the measured able damage locations are ranked according to the response characteristics and the corresponding re- posterior probabilities of the corresponding model classes. The most probable model class will be in- 3 OPTIMAL SENSOR LOCATION BASED ON dicative of the substructure that is damaged, while INFORMATION ENTROPY the probability distribution of the model parameters of the corresponding most probable model class will The information entropy H(δ , D ) (Papadimitriou et be indicative of the severity of damage in the identi- al. 2000), which is a unique scalar measure of the fied damaged substructure. uncertainty in the estimate of the structural parame- Using Bayes’ theorem, the posterior probabili- ters q , is used for optimizing the sensor configura- ties of the various model classes given the data D is tion in the structure for identifying a model class Μ . The information entropy depends on the available p( D |Μi ) P ( Μ i ) δ P(Μ | D ) = data D and the sensor configuration vector . It i d has be shown (Papadimitriou 2004) that for a large number of measured data, i.e. as ND N  , the where p( D |Μ i ) is the probability of observing the following asymptotic result holds for the informa- data from the model class Μ i , P(Μ i ) is the prior tion entropy for a model class Μ probability of the model class Μ i , while d is selected so that the sum of all model probabilities equals ˆ1 1 ˆ HΜ(δ , D ): H Μ ( δ ;q ,sˆ )= N ln(2 p ) - ln[det h Μ ( q , s ˆ ; δ )] to one. Assuming there is no prior preference as to 2q 2 what class of models we choose, we may set that

P(Μ )= 1/ in . ˆ ˆ i m where q� q(δ ,D ) arg min J ( q ; D ) is the optimal The following asymptotic approximation has value of the parameter q set that minimizes the been introduced to give a useful and insightful esti- q measure of fit JΜ (q ; D ) J ( q ; D ) given in or for a mate of the integral involved in p( D |Μ ) in (Pa- ˆ ˆ model class Μ , and hΜ(q , sˆ ) hδ Μ ( q , s ˆ ; ) is an padimitriou and Katafygiotis 2004) N N q q positive definite matrix defined by and - ND N N / 2 ˆ ˆ ˆ asymptotically approximated by (Papadimitriou q p (q |Μ ) [JΜ ( q , s ))] p( D |Μ ) ~( 2p) q ˆ 2004) dethΜ (q , sˆ ) ˆ ˆ where, for uniform prior distribution of the paramet- hδΜ(q , s Qˆ ; ) δ: Μ ( , q , s ˆ ) ers q in a model class Μ , ˆ is the value that min- ˆ ˆ q imizes JΜ (q , sˆ ) , and hΜ (q , sˆ ) is defined by as ND N  . For response time history data, substituting into and considering the limiting case ND N  , the ˆ ˆ 蜒 T - ND N hΜ (q , s )=- ln[J ( q ; D )] ˆ q q q= q resulting matrix Qδ( ,q ) appearing in simplifies to a positive semi-definite matrix of the form =[抖 / ,⋯ , 抖 / ]T in which q1 qN is the usual Nd q q N 1 gradient vector with respect to the parameter set , ˆD (j ) ˆ q QδΜ( ,q , sˆ )= dj P Μ ( q ) 2 2 ˆ 2 and sˆ is the optimal prediction error variance. The j=1 s j approximate estimate is unreliable when the optimal u ˆ is outside the region Q ={q : q q } of varia- that contains the information about the values of the q u tion of q , where q are the values of q at the un- parameters q based on the data from all measured damaged condition. Alternatively, one can use im- positions specified in δ , while the optimal predic- portance sampling method to compute the integral 2 ˆ 2 = ˆ tion error variances sˆ are given by sjJ j ( q ; D ) . involved in estimating p( D |Μ ) in (Papadimitriou The matrix and Katafygiotis 2004). ˆ N The matrix hΜ (q , sˆ ) in the denominator of is D P(j ) ( )= 蜒 x ( k ; ) T x ( k ; ) the Fisher information matrix quantifying the uncer- Μ qqj q q j q tainty in the model parameters of the model class Μ k=1 (Papadimitriou et al. 2000). The result in suggests that the selection of the optimal model class among is a positive semi-definite matrix containing the in- Μ⋯ Μ formation about the values of the parameters the m model classes 1 m depends on the fit q each model class provides to the measured data as based on the data from one sensor placed at the j -th 2 ˆ well as the uncertainty in the parameter values of DOF. The prediction error sˆ j= J j (q ; D ) in is com- 2 2 2 ˆ each model class. The sensor locations are known to puted from sˆ j=s1 + s 2 g j (q ) , where the first term ˆ accounts for constant measurement error and the affect the value of the information matrix hΜ (q , sˆ ) and the uncertainty in the model parameters of each second term accounts for model error that depends ˆ model class. An optimal sensor configuration strate- on the strength g j (q ) of the response characteristics 2 2 gy should provide informative measurements for the with the values of s1 and s2 giving the relative size Μ⋯ Μ of measurement and model errors. multiple model classes 1 m. For modal data, the resulting matrix QδΜ ( ,q ) dices for all m model classes is formulated as a mul- simplifies to a positive semi-definite matrix given by ti-objective optimization problem stated as follows. Find the values of the discrete-valued parameter set

轾 T N thatT simultaneously minimizes the objectives (Pa- N m 蜒 ( ) ( ) d 蜒 L(d ) L ( ) D犏 qw rq q w r q q0fjrpadimitriou q q 0 f jr 2005). q QδΜ ( , ) = + j q 邋犏 2 2 2 d 2 2 2 2r=1犏 s1+ s 2 wr (q ) j = 1 s+ s L( ) / N 臌 1 2 0fr = q 0 J (d ) (J1 ( d ),..., J m ( d )) J( ),..., J ( ) containing the information about the values of the For conflicting objectives 1 dm d , there is model parameters q based on the modal data from no single optimal solution, but rather a set of alterna- all sensors placed in the structure. tive solutions, which are optimal in the sense that no The asymptotic approximation of the informa- other solutions in the search space are superior to tion entropy is useful in the experimental stage of them when all objectives are considered. Such alter- designing an optimal sensor configuration. Specifi- native solutions, trading-off the information entropy cally, the information entropy (7) for a model class values for different model classes, are known in ˆ multi-objective optimization as Pareto optimal solu- Μ is completely defined by the optimal value q of the model parameters and the optimal prediction er- tions. An advantage of the multi-objective identifica- 2 ˆ tion methodology is that all admissible solutions are ror sˆ j= J j (q ; D ) , j=1,⋯ , N0 , expected for a set of test data, while the time history details of the obtained which constitute model trade-offs in reduc- measured data do not enter explicitly the formula- ing the information entropies for each model class. tion. The optimal sensor configuration is selected as These solutions are considered optimal in the sense the one that minimizes the information entropy (Pa- that the corresponding information entropy for one model class cannot be improved without deteriorat- padimitriou et al. 2000) with respect to the set of N0 measurable DOFs. However, in the initial stage of ing the information entropy for another model class. designing the experiment the data are not available, The optimal points along the Pareto trade-off front and thus an estimate of the optimal model parame- provide detailed information about the effectiveness ˆ of the sensor configuration for each model class. ters q and sˆ cannot be obtained from analysis. In practice, useful designs can be obtained by taking An exhaustive search over all sensor configura- ˆ tions for the computation of the optimal sensor con- the optimal model parameters q and sˆ to have some nominal values chosen by the designer to be figuration is extremely time consuming and in most representative of the system. cases prohibitive. Alternative, genetic algorithms are well suited for performing the multi-objective optimization involving discrete variables. In particular, 4 OPTIMAL SENSOR LOCATIONS FOR MULTI- the strength Pareto evolution algorithm (Zitzler and PLE MODEL CLASSES Thiele 1999) based on genetic algorithms is most suitable for solving the resulting discrete optimiza- The design of optimal sensor configurations for pro- tion problem and providing near optimal solutions viding informative measurements for multiple model (Papadimitriou 2005). Μ Μ A more systematic and computationally very ef- classes 1,..., m is next addressed. Let ficient approach for obtaining a good approximation Ji(d ) IEI i ( d ) be the effectiveness of a sensor con- of the Pareto optimal front and the corresponding figuration d for the i th model class Μi , where Pareto optimal sensor configurations for a fixed IEIi (d ) is the information entropy index given by number of N0 sensors is to use a sequential sensor placement (SSP) approach (Papadimitriou 2005), ex- H ( ) - H id i( d i, best ) tending the SSP algorithm (Papadimitriou 2004) to IEIi (d) = Hi(d i, worst )- H i ( d i , best ) handle Pareto optimal solutions. The total number of vector function evaluations using the extended SSP ˆ algorithm is infinitesimally small compared to the with Hi (d )篸 HΜ ( ; q ,sˆ ) . The optimal sensor con- i number of vector function evaluations required in an figuration for the model class Μi is selected as the one that minimizes the information entropy index exhaustive search method. Numerical applications (Papadimitriou 2005) show that the Pareto front con- J i (d ) . In , di, best is the optimal sensor configuration structed by this heuristic algorithm, in most cases and di, worst is the worst sensor configuration for the i examined, coincides with, or is very close to, the ex- th model class. The values of IEIi (d) range from zero to one. The most effective configuration corre- act Pareto front. In all cases, the extended SSP algorithm outperforms, in terms of accuracy and compu- sponds to value of IEIi (d) equal to zero, while the least effective configuration corresponds to value of tational time, available discrete multi-objective optimization algorithms such as the strength Pareto evo- IEIi (d) equal to one. The problem of identifying the optimal sensor lution algorithm based on genetic algorithms (Har- locations that minimize the information entropy in- alampidis et al. 2005). 4 ILLUSTRATIVE EXAMPLE the damage scenario considered. The case of N0 = 3 sensors and the case of two and three contributing A numerical example is given to illustrate the modes is considered. The measurement locations for theoretical developments. Consider a structure repre- N0 = 3 sensors are taken in all cases to be at the op- sented by a Nd -DOF chain-like spring-mass model timal locations {1, 9, 10} DOFs of the chain as well with one end of the chain fixed at the base and the as the alternative non-optimal locations {8, 9, 10} other end free. The nominal values of the stiffness and {6, 8, 10}. The results are computed using both and mass of each link in the chain are chosen to be asymptotic (first row) and importance sampling ki = k0 and mi = m0 , i=1,⋯ , Nd , respectively. The (second row) method. nominal model is considered to be the model of the It is seen in Table 1 that the effectiveness of the structure at its undamaged condition. methodology in predicting the location of damage A family of model classes depends on the number of modes and the location of {M12 , M 34 , M 56 , M 78 , M 9,10 } is considered so that sensors. Specifically, comparing the probabilities in each model class M ij is parameterized by two pa- Table 1 for all model classes for the case of three rameters that account for the stiffness values of two modes ( m = 3 ) and for the optimal sensor configu- out of the ten links. In the selection of the model ration {1,9,10}, it is seen that the methodology cor- classes, it was assumed for demonstration purposes rectly predicts the location of damage since it gives a that damage is localized in two adjacent links. For probability of one to model class M12 and zero each model class the stiffness and mass properties of probability to all other model classes. Even for the links that are not parameterized equal the nomi- m = 2 modes, the methodology based on impor- nal values of the undamaged structure. tance sampling estimates of the probabilities favors In order to show the influence of the sensor con- model class M12 with high probability 0.99. Howev- figuration on the prediction effectiveness of the er, in the case of m = 2 modes, the asymptotic ap- methodology, the case of N0 = 3 sensors is consid- proximation provides incorrect predictions of the ered. The total number of sensor configurations for relative probabilities of the model classes. This is this problem is 120. Among all these configurations, because the estimate of the probability of the model 64 are Pareto optimal sensor configurations for the 5 class M 78 is incorrect due to the fact that the optimal u model classes considered. Among all Pareto optimal q is well outside the range q[0, q ]. sensor configurations, the configuration {1, 9, 10} gives no preference to any model class by providing Table 1. Asymptotic (first row) and importance sampling (sec- almost equally informative data for all model classes ond row) estimates of the probabilities of the model classes. simultaneously, in the sense that Ji (d ) J min for all Sensor Location Sensor Location Sensor Location i =1,⋯ ,5 with the minimum value of Jmin = 0.638. {1,9,10} {8,9,10} {6,8,10} Among the remaining 63 Pareto optimal sensor con- Model figurations, there are configurations d that provide m = 2 m = 3 m = 2 m = 3 m = 2 m = 3 more informative data for one or more model classes Class J ( )< 0.638 0 1.0 0 0.64 0 0.75 with i d , but simultaneously provide less M12 informative data for other model classes in the fami- 0.99 1.0 0.62 0.64 0.62 0.73 ly with J ( )> 0.638 . The Pareto optimal sensor 0 0 0.55 0 0.99 0 i d M34 configuration {1, 9, 10} is found to provide the most 0 0 0 0 0 0 informative data considering all model classes si- 0.01 0 0 0.36 0 0.25 M56 multaneously. 0.01 0 0.36 0.36 0.38 0.27 1.0 0 0.45 0 0.01 0 A damage scenario is considered that corre- M sponds to 40% stiffness reduction in the first link. 78 0 0 0.02 0 0 0 0 0 0 0 0 0 Simulated measured modal data are generated by the M model with 40% reduced stiffness at the first link. 910 0 0 0 0 0 0 To simulate the effects of measurement noise and modeling error, 2% and 5% noise are respectively Using the other two non-optimal sensor config- added to the modal frequencies and modeshapes urations {8, 9, 10} and {6, 8, 10}, the methodology simulated by the damaged models. It is expected that correctly predicts the location of damage but with the application of the methodology should yield as significantly lower probability than the one obtained most probable model class the M12 , with the value for the case of sensor configurations {1, 9, 10}. This of one of the two parameters predicting the severity is due to the lack of significant information for up- of damage in the damaged link. The probability of dating model class M12 provided by the non-optimal each model class is obtained by . A non-informative sensor configurations. Besides favoring model class prior probability distribution for the parameters of M12 with probability 0.64 (for sensor configuration each model is considered. {8, 9, 10}) or 0.73 (for sensor configuration {6, 8, The results for the probability P( Mij | D N ) for 10}), the methodology also favors the model class all model classes considered are given in Table 1 for M 56 with probability 0.36 or 0.27, respectively. This is indicative of the fact that the reliability of the pre- Social Fund) and 25% from the Greek Ministry of dictions depend on the location of sensors in the Development (General Secretariat of Research and structure which affects the amount of information Technology) and the private sector, under grant 03- contained in the data for updating each model class ΕΔ-524 (PENED 2003). in the family of model classes. Note that the reliability of predictions deteriorates for the case of m = 2 modes. REFERENCES It should be noted that for all three sensor configurations and for N = 3 modes, the asymptotic Beck, J. L. & Katafygiotis, L. S. 1998. Updating Models and and importance sampling estimates give qualitative- their Uncertainties - Bayesian Statistical Framework. ly similar prediction for the location of damage. 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ACKNOWLEDGEMENTS

The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (CityU 114706). Also, it was co- funded 75% from the European Union (European