Solving the Damage Localization Problem in Structural Health Monitoring Using Techniques in Pattern Classification

Transcription

1 Solving the Daage Localization Proble in Structural Health Monitoring Using Techniques in Pattern Classification CS 9 Final Project Due Dec. 4, 007 Hae Young Noh, Allen Cheung, Daxia Ge

2 Introduction Structural health onitoring (SHM) for civil engineering applications involves the use of sensing networks in order to diagnose daage in large scale civil structures such as buildings or bridges. Health onitoring will result in large savings in aintenance costs as well as iproving life safety by providing a real-tie updating of a structure s integrity. Recent events such as the bridge collapse in Minneapolis as well as aging infrastructure illustrate the need to iprove structural onitoring. Recent research has shifted towards using statistical algoriths and tie series odeling to detect daage []. Statistical ethods are advantageous, because they do not require the user to create a structural, usually finite eleent, odel for the whole structure. This results in significant savings in power consuption as well as data transission, aking the ideal for use in low power wireless networks. Interest in using wireless networks for SHM has increased due to the lower cost and ease of installation []. Daage diagnosis is divided into three categories: daage identification, easureent of daage extent, and daage localization. Extensive work has been done in the area of daage identification and localization. In particular, Nair et al. (006) show that an algorith based on autoregressive tie series odeling of a structure s acceleration response to abient vibrations is capable of detecting and easuring the extent of daage on a structure. However, the daage localization proble reains unsolved. Localization is iportant, because it will allow responders to strealine efforts in repair and aintenance. This paper exaines the use of pattern classification techniques in achine learning to solve the daage localization proble. Following, Nair et al. (006), autoregressive coefficients fro a structure s acceleration response are selected as the feature vector. Data is collected by siulation using the ASCE Phase I Benchark Structure [3]. It is shown that using established pattern classification techniques, it is possible to localize daage in a structure. Further work is shown in exaining the optial nuber of paraeters and nuber of training exaples for this proble. Tie Series Modeling by Autoregressive Coefficients Nair et al. (006) show that the autoregressive (AR) coefficients fro a structure s accelerations under abient vibrations are directly related to the structure s natural frequency and by extension its stiffness []. The autoregressive tie series odel can be represented as a linear difference equation with a transfer function, with the locations of the poles given by the AR coefficients; thus it follows that AR coefficient values will be a function of the structural stiffness. Thus changes to a structure s stiffness atrix as a result of peranent daage will change the structure s AR coefficients. For a ore coplete derivation of the AR algorith, see Nair et al. (006). The AR algorith is perfored according to the following steps as was previously done by Sohn et al. (00) and Nair et al. (006): The acceleration tie history is noralized and standardized according to the following equation: (x(t) μ) x~ (t) = σ ()

3 where x(t) is the acceleration tie history at a point t in tie, μ and σ are the ean and standard deviation of the tie history, and x (t) is the noralized and standardized tie history. Let x ĩ (t) be the noralized and standardized acceleration signal fro the i th sensor. In order to increase the nuber of features, x i(t) is divided into N chunks. Each chunk is an independent feature. More specifically, Nair et al. (006) show that if the acceleration tie history can be shown to be stationary, then the acceleration signal of each chunk will be independent and identically distributed. Although this has been done, it will not be shown here for brevity. The AR odel is given by the following equation: ~ x ij p () t α ~ x ( t k) + ε ( t) = k = ijk ij where x ĩj (t) is the noralized acceleration signal of the j th chunk of the i th sensor, α ijk is the k th AR coefficient fro the jth chunk of the ith sensor, p is the AR odel order, and ε ij (t) is the residual ter. The Burg algorith (Brockwell and Davis, 003) is used to estiate the AR coefficients. In this study a odel order of 5 and a chunk size of 4000 are used. The optial AR order p and optial chunk size N can be deterined according to ethods proposed by Nair et al. (006). Again for brevity, they will not be discussed. ASCE Phase I Benchark Structure The ASCE Phase I Benchark Structure is a standardized siulation proble for the developent of structural health onitoring algoriths for civil structures. The benchark structure is a Matlab based siulation of a structure s response to abient vibrations. Daage can be specified according to 6 predeterined daage patterns or user defined on any cobination of beas, coluns, and braces. Output is given in the for of 6 acceleration tie histories, representing acceleration fro 8 different sensors in the x and y directions. Extensive data can be gathered by specifying the rando seed for force generation. The odel is based off of an existing structure which was constructed and tested at the University of British Colubia. The structure is a two bay, four story, rectangular steel braced frae. It is odeled as a 0 degree of freedo syste with various loading options. The Phase I Benchark is chosen instead of a ore recent benchark, because the ability to assign user defined daage patterns is critical in order to build a suitable training and test data set required for the ipleentation of pattern classification algoriths. Pattern Classification Once the AR algorith is perfored on the acceleration tie histories, the AR coefficients are assebled into a feature vector. Previous work done by Nair et al. (006) and confired by this study show that the first AR coefficient is sufficient to pick up changes in structural stiffness resulting fro daage. For this reason the feature vector X is chosen to be a vector of the first AR coefficient fro all the sensors available. ij () X j α j α j =... α nj (3)

4 where n is the nuber of sensors, and j is the index of the current chunk. For the benchark structure n = 6, representing 8 acceleroeters each recording acceleration in the x and y directions. The training set size is equal to the nuber of chunks per run N ultiplied by the nuber of runs. A run is defined as one single output gathered fro siulation by subjecting the structure to one ground otion and subjecting it to one daage case. Daage location is discretized according to the floor on which the daage has occurred. This results in 5 possible classifications for daage location: undaaged, st floor, nd floor, 3 rd floor, or 4 th floor. Thus this becoes a ulti-category classification proble. One way to solve this proble is to use binary classification ethods and ipleent the in a one-versus-all (va) or one-versus-one (v) approach (Duan et al. 003). In this study two binary classification ethods are used: regularized logistic regression using Newton s ethod to solve the gradient ascent proble and support vector achines (SVM) using sequential inial optiization (SMO) with a linear kernel. The SMO algorith is used as derived by Platt (998). Gradient ascent attepts to axiize the following log-likelihood function: θ (4) MAP = argax [ y log( hθ ( x ) + ( y ) log( hθ ( x )] λ θ θ i= where θ MAP is the estiate of the odel paraeters, y (i) is the ith class label, either 0 or, x (i) is the ith feature vector, and h θ (x) is the signoid function given by: λ θ is the Bayesian regularization ter. θ is assued to be ~N(0, τ I). λ is chosen using cross-validation. The log-likelihood function is axiized using the Newton-Raphson ethod. ( t+ ) ( t ) θ = θ H θl (7) The paraeters θ are updated according to the previous equation until convergence has occurred or until soe specified axiu nuber of iterations has taken place. l is the log-likelihood function, H is the Hessian atrix of l with respect to θ and is the gradient of the log-likelihood function. l θ h ( x) T ( x) = + e θ θ λ = τ (5) (6) SVM attepts to solve the Dual Proble. The proble siplifies to a axiization proble given by the following expression: such that: ax α i= α i i= j= α i y ( i) ( j ) ( i) ( j ) 0 y α α i j x, x (8) (9) 3

5 This axiization proble is solved using SMO as derived by Platt (998). A description of that algorith is beyond the scope of this paper. For further details see Platt (998). Both the va and v approach for ulti-classification are used with a winnertakes-all (WTA) and a ax-wins voting (MWV) strategy (Duan et al. 003). In addition, each algorith was perfored with the diension of the feature vector reduced using Principal Coponents Analysis (PCA). The feature diension n = 6 for the benchark structure is anageable 8 sensors easuring acceleration in the x and y directions. However, for a larger structure such as a suspension bridge or skyscraper, n ay grow into the thousands, especially if the z direction of acceleration is also considered. PCA provides a way to ake coputations anageable. For this study, the diension of the feature vector was reduced to the first two principal coponents. The following daage patterns were introduced to the benchark structure. Table Daage Patterns and Class Labels Daage Pattern Class Label Undaaged All st Floor Braces Daaged All st Floor Beas Daaged All nd Floor Braces Daaged 3 All nd Floor Beas Daaged 3 All 3 rd Floor Braces Daaged 4 All 3 rd Floor Beas Daaged 4 All 4 th Floor Braces Daaged 5 All 4 th Floor Beas Daaged 5 i= y α = 0 0 runs are generated for each daage state, each with different rando seeds. Each run is divided into 0 chunks according to the algorith described above. Therefore the size of the total data set is 900. Two daage states are considered daage to beas and daage to braces in order to deterine the feasibility of localizing ultiple types of daage even though the daage is labeled only according to location. It was found in the analysis that the daage localization algorith trained on brace daage data alone was incapable of localizing bea daage. The data are labeled strictly according to which floor the daage took place on. All algoriths were trained on 70% of the data, with the other 30% reserved for testing. The size of the training set is 630. The size of the test set is 70. Through cross validation, it was found that an optial value of λ for gradient ascent is 0.0 and an optial value of C for SVM is 50. The results of each algorith are copared in Table. Figure shows the distribution of features when plotted according to their first two principal coponents. A test cased was deterined to be correctly assessed if the algorith predicted the floor on which the daage occurred (or no daage). An incorrect assessent was recorded as an error. i (0) 4

6 Table Test Error Results for Selected Algoriths Gradient Ascent SMO v MWV.% 0.37% va WTA.6% 0% v MWV PCA 64.8% 49.6% va WTA PCA 65.% 6.6% Figure Distribution of Features According to First Two Principal Coponents The pattern classification algoriths used are able to predict with high levels of accuracy the location of daage in the benchark structure. As expected, PCA increases the aount of test error, because inforation is lost. Nevertheless, even with the diensionality of the feature vector reduced to, the v SMO algorith is able to predict with 50% accuracy the location of daage in the structure. Conclusion A pattern classification approach using supervised learning shows excellent results when tested on siulated data. SVM-based ulti-classifier algoriths are able to achieve alost 00% success rate in identifying the location of daage. The advantage of this approach is that once a training data set has been built, testing is relatively siple and low in coputational coplexity. The proble with ipleenting this approach in practice to onitor existing structures is the lack of and difficulty in acquiring a large enough training set. Because acquiring a training set experientally would be prohibitively expensive, future research work should exaine whether a training set created using siulated data can be used to predict the location of actual daage on an experientally tested structure. In addition to this, future research ay consider less sever daage states, other types of daage, other ways of discretizing daage location, and whether training data for one type of structure can be applied successfully to siilar structures. However, despite the difficulties with applying this approach in practice, supervised learning approaches offer a great deal of potential in solving the daage localization proble. 5

7 References Nair, K.K., Kireidjian and K.H. Law. (006). Tie Series Based Daage Detection and Localization Algorith with Application to the ASCE benchark Structure, Journal of Sound and Vibration 9, Sohn, H., Farrar, C.R., Hunter, H.F. and Worden, K., (00), Applying the LANL Statistical Pattern Recognition Paradig for Structural Health Monitoring to Data fro a Surface-Effect Fast Patrol Boat, Los Alaos National Laboratory Report LA-376-MS, Los Alaos National Laboratory, Los Alaos, NM Brockwell, R. J. and Davis, R. A. (003). Introduction to Tie Series and Forecasting, Springer, nd Ed., 456p. Platt, John. Fast Training of Support Vector Machines using Sequential Minial Optiization, in Advances in Kernel Methods Support Vector Learning, B. Scholkopf, C. Burges, A. Sola, eds., MIT Press (998) Duan, Kaibo, S. Sathiya Keerthi, Wei Chu, Shirish Krishnaj Shevade, and Aun Neow Poo. (003). Multi-Category Classification by Soft-Max Cobination of Binary Classifiers, Multiple Classifier Systes, pp