T HE ATRIUM, SOUTHERN GATE, CHICHESTER, WEST SUSSEX P019 8SQ

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1 T HE ATRIUM, SOUTHERN GATE, CHICHESTER, WEST SUSSEX P0 8SQ ***IMMEDIATE RESPONSE REQUIRED*** Your article may be published online via Wiley's EarlyView service ( shortly after receipt of corrections. EarlyView is Wiley's online publication of individual articles in full-text HTML and/or pdf format before release of the compiled print issue of the journal. Articles posted online in EarlyView are peer-reviewed, copy-edited, author-corrected, and fully citable via the article DOI (for further information, visit EarlyView means you benefit from the best of two worlds - fast online availability as well as traditional, issue-based archiving. READ PROOFS CAREFULLY Please follow these instructions to avoid delay of publication This will be your only chance to review these proofs. Please note that once your corrected article is posted online, it is considered legally published, and cannot be removed from the Web site for further corrections. Please note that the volume and page numbers shown on the proofs are for position only. ANSWER ALL QUERIES ON PROOFS (Queries for you to answer are attached as the last page of your proof.) List all corrections and send back via , or mark all corrections directly on the proofs and send the scanned copy via . Please do not send corrections by fax or in the post. corrections to: wileyjournals@macmillan-india.co.in CHECK FIGURES AND TABLES CAREFULLY Check size, numbering, and orientation of figures. All images in the PDF are downsampled (reduced to lower resolution and file size) to facilitate Internet delivery. These images will appear at higher resolution and sharpness in the printed article. Review figure legends to ensure that they are complete. Check all tables. Review layout, title, and footnotes. COMPLETE CTA (if you have not already signed one) Please send a scanned copy with your proofs and post your completed original form to the address detailed in the covering . We cannot publish your paper until we receive the original signed form. OFFPRINTS 2 complimentary offprints of your article will be dispatched on publication. Please ensure that the correspondence address on your proofs is correct for dispatch of the offprints. If your delivery address has changed, please inform the production contact to the journal - details in the covering . Please allow six weeks for delivery. Additional reprint and journal issue purchases Additional paper reprints (minimum quantity 00 copies) are available on publication to contributors. Quotations may be requested from mailto:author_reprints@wiley.co.uk. Orders for additional paper reprints may be placed in advance in order to ensure that they are fulfilled in a timely manner on publication of the article in question. Please note that offprints and reprints will be dispatched under separate cover. PDF files of individual articles may be purchased for personal use for $2 via Wiley s Pay-Per-View service (see Please note that regardless of the form in which they are acquired, reprints should not be resold, nor further disseminated in electronic or print form, nor deployed in part or in whole in any marketing, promotional or educational contexts without further discussion with Wiley. Permissions requests should be directed to mailto:permreq@wiley.co.uk Lead authors are cordially invited to remind their co-authors that the reprint opportunities detailed above are also available to them. If you wish to purchase print copies of the issue in which your article appears, please contact our Journals Fulfilment Department mailto:cs-journals@wiley.co.uk when you receive your complimentary offprints or when your article is published online in an issue. Please quote the Volume/Issue in which your article appears.

2 STC : 22 PROD.TYPE: COM ED: VIJAYA pp.^8 (col. g.: NIL) B2 PAGN: KG.THILAKAM SCAN: STRUCTURAL CONTROL AND HEALTH MONITORING Published online in Wiley InterScience ( DOI: 0.002/stc.22 A novel health assessment technique with minimum information Hasan Katkhuda,2,},} and Achintya Haldar 2, *,y,z The Hashemite University, Zarqa, Jordan 2 Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ, U.S.A. SUMMARY A novel structural health assessment method is proposed for detecting defects at the element level using only minimum measured response information considering imperfect mathematical model representing the structure, various sources of uncertainty in the mathematical model, and uncertainty in the measured response information. It is a linear time domain finite element-based system identification procedure capable of detecting defects at the local element level using only limited response measurements and without using any information on the exciting dynamic forces. The technique is a combination of the modified iterative least-squares technique with unknown input excitation (MILS-UI) proposed earlier by the research team at the University of Arizona and the extended Kalman filter with a weighted global iteration (EKF-WGI) procedures to address different sources of uncertainty in the problem. The authors denote the new method as GILS-EKF-UI. To implement the concept, a two-stage substructure approach is used. In the first stage, a substructure is selected that satisfies all the requirements for the MILS-UI procedure. This provides information on the initial state vector and the input excitation required to implement any EKF-based procedure. In the second stage, the EKF-WGI is applied to identify the whole structure. This way the whole structure, defect free or defective, can be identified with limited response measurements in the presence of uncertainty and without using excitation information. The theoretical concept behind this novel method is presented in this paper. Copyright # 200 John Wiley & Sons, Ltd. KEY WORDS: damage assessment; finite elements; Kalman filter; system identification under uncertainty; structural health assessment *Correspondence to: Achintya Haldar, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ, U.S.A. y haldar@u.arizona.edu z Professor and da Vinci Fellow. } Assistant Professor. } Doctoral Student. 4 Contract/grant sponsor: University of Arizona Foundation; contract/grant number: 4848 Copyright # 200 John Wiley & Sons, Ltd. Received January 200 Revised June 200 Accepted 0 July 200 stc 22

3 STC : 22 2 H. KATKHUDA AND A. HALDAR INTRODUCTION Health monitoring of existing structures has recently become an important and challenging issue to the engineering profession and generated multi-disciplinary research interest. Structures deteriorate as they age and can also suffer damage due to natural events like major earthquakes or high winds. Man-made events like explosions can also cause damage. In all cases, the performance of structures is expected to change to reflect the amount of degradation they have suffered. Objective health assessment is an integral part of the performance-based design concept now being promoted in the profession. Visual inspection on a regular basis or on an emergency basis just after a major event is generally suggested or recommended in the design guidelines. However, defects invisible to the naked eye cannot be detected. Visual inspection may be impractical in some cases due to accessibility problems and may be ineffective in some other cases. The outcome of inspections of visually detectable defects may depend on many subjective factors including the experience of the inspectors, the type of structure being inspected, and the location of the defects. An objective, efficient, economical, and robust nondestructive inspection (NDI) procedure is urgently needed for the structural health assessment (SHA) purpose. Global dynamic structural responses can be used for SHA since they depend on the current health of all structural elements. The state of the structural system can be identified using a system identification (SI) technique using the information on the exciting force and the dynamic responses caused by the force. The current state or health of all structural elements can be established by tracking the changes in the dynamic responses or signature. This is a multidisciplinary research area. The available literature on the related topics is very extensive [, 2]. It is not practical to cite majority of them in a technical paper. Only some of them specifically related to the proposed study are cited in this paper. The potential of SI as an NDI technique for SHA in the presence of several sources of uncertainty has received attention from the research community only recently. In most cases in the past, verifications of the SI techniques were conducted using computer-generated noise-free and artificially introduced noise-contaminated response information. When a method failed to identify, the presence of excessive amount of noise in the response information was considered to be the main reason. In fact, Maybeck [] commented that consideration of a deterministic system without considering imperfection in the mathematical model, the presence of various sources of uncertainty in the mathematical model, and noise in the measured response information, might make the SI-based approach invalid. However, the potential of SI-based SHA technique is being advocated by the research community in many international conferences. Most of the currently available SI techniques are based on the frequency domain approach [4 ]. There are many advantages to the frequency domain approaches. The modal information can be expressed in countable form in terms of frequencies and mode shape vectors for comparison, and changes in them can indicate the current health of a structure. Since modal information represents global information, there may be an averaging effect for the noise or error in the response measurements. It is not necessary to manipulate a large amount of response data in this approach. However, the changes in the frequencies and mode shapes can help to establish the altered or defective state of a structure, i.e. whether the structure is defective in the global sense, but the locations of defects at the element level cannot be easily detected with this approach. Furthermore, the frequencies may not change significantly even in the presence of Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

4 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION major defects, indicating low sensitivity for defect assessment purposes. The modal information loses its effectiveness as a damage indicator if only the first few modes are used. Higher-order modes are difficult to evaluate, particularly for large structural systems, and there are no guidelines on how many modes are required for effective defect identification. The research team at the University of Arizona and some other scholars commented that time domain approaches would be more appropriate for identifying defects at the local element level [, 2 ]. To address this recommendation, a novel time domain SI technique is proposed in this paper. Available time domain SI approaches with potential for SHA can be grouped in to leastsquares estimation [6 8], extended Kalman Filter (EKF) [ 22], H Filter [2], Monte Carlo filter [24], sequential nonlinear least-squares estimation [22], and other methods. All these methods require input excitation information and will not satisfy the basic objective of the proposed study, i.e. SI without input excitation information. Yang et al. [22] in 2006 commented that To date, however, no analytical or recursive solution has been obtained for the EKF approach when the external excitations are not measured. Accepting the proposition that a time domain SI approach will be more appropriate, the implementation potential of the method to identify defects in real structures in the presence of various sources of uncertainty needs to be investigated, particularly addressing the comments made by Maybeck []. Furthermore, the excitation force is very difficult to measure outside the highly controlled laboratory environment. The measured excitation force could be so noisy that the overall SI concept might not be applicable. Furthermore, the information on the excitation force may not be available after natural or man-made events. The application potential of a time domain SI approach for detecting defects at the element level could be significantly improved if a structural system can be identified using only response information. The task is expected to be very challenging since two of the three elements of the SI concept (input excitation, the system to be identified, and the output response) will be unknown. Some of the SI approaches without input excitation information are the KF-WGI with running load [2], stochastic-adaptive techniques [26], free-decay curve analysis [2 0], stochastic approach [, 2], random decrement [2, ], auto-regressive moving average with stochastic input [6], eigenspace approach [], auto-regressive moving average vector [8], and iterative least-square schemes [, 40]. Most of these methods need modal properties, have limitations on output information, cannot identify at the element level, and will be unable to satisfy the objectives of the study [4]. Wang and Haldar [4] established a conceptual framework for a SI approach without input excitation information. They called it the iterative least square with unknown input (ILS-UI) method. They used viscous damping in the governing equation of motion. They identified sheartype buildings (buildings are assumed to deflect under shear force only), the simplest mathematical representation of complicated structural systems. The total mass of the structure is lumped at the floor levels, assigning one dynamic degree of freedom (DDOF) for each floor corresponding to the horizontal displacement; the girders/floors are assumed to be infinitely rigid compared with the columns; all the columns in a floor are represented by one column, the deformation of the structure is considered to be independent of the axial force present in the columns, rotational DDOFs are completely ignored, and the mass and stiffness matrixes are considered to be diagonal. Assuming the mass matrix is known (a very common assumption reported in the literature), for an n story shear-type building, the method identifies n numbers of stiffness and damping parameters by identifying a total of 2n parameters. Wang and Haldar, using computer-generated response information, demonstrated that the method is robust and Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

5 STC : 22 4 H. KATKHUDA AND A. HALDAR accurate and can identify a system even in the presence of artificially added large amount of noise in the responses. To improve the efficiency of the identification process, i.e. instead of identifying a total of 2n parameters, Ling and Haldar [42] proposed using Rayleightype damping instead of viscous damping in the governing equation, i.e. the damping is proportional to mass and stiffness. In this formulation, only ðn þ 2Þ parameters need to be identified. This approach significantly improves the efficiency of the formulation, particularly for large structural systems. They called it the modified ILS-UI method or MILS-UI. If a structure is represented by finite elements, the MILS-UI method adds simplicity to the formulation. Wang and Haldar and Ling and Haldar established the viability of the concept, i.e. a system can be identified in the computer environment using only noise-contaminated response information. The major drawback of the MILS-UI approach is that the response information must be available at all DDOFs. For real structural systems, the required response information is not expected to be available at all DDOFs. To increase the application potential, the MILS-UI method needs to be extended so that structural systems can be identified with only limited response information in the presence of several sources of uncertainty. This is the subject of this paper. A Kalman filter-based approach will be very appealing when response information is limited [4, 44] and the problem contains several sources of uncertainty as discussed by Maybeck []. It will provide an ideal platform for the proposed method. The Kalman filter is an optimal recursive data processing algorithm which processes the available response measurements regardless of their precision. To increase the efficiency of the optimization algorithm, Hoshiya and Saito [] proposed the extended Kalman filter with weighted global iteration (EKF-WGI) method. The algorithm uses the prior knowledge about the system, the information on the input excitation, and limited response information to produce an estimation of the desired variables by statistically minimizing the error. Most of the past works on SI using the concept were limited to identifying very small systems containing not more than 0 DDOFs [, 2, 4 ]. To implement the algorithm, the information on the state vector and the excitation force, fðtþ; must be available. The use of the Kalman filter without any information on input excitation is not possible. At the same time, the primary objective of the proposed approach is to identify a structure without using any information on the input excitation. The two requirements contradict each other. Wang and Haldar [] attempted to address these issues by identifying shear-type buildings. They called it the iterative least-square extended Kalman filter with unknown input (ILS-EKF-UI) method. It completely ignored the rotational DDOFs and viscous damping was used in the formulation. Furthermore, it cannot identify a real structure at the local element level. The ILS-EKF-UI method will not satisfy the objectives of this study. To identify different types of complicated realistic structural systems, the authors proposed a twostage Kalman filter-based approach and it is the main topic of this paper. Two stages are required to satisfy all the requirements of the Kalman filter technique. They called it the generalized ILS-EKF-UI method. Using the GILS-EKF-UI method, the stiffness parameter of all the elements in the finite element representation can be identified. The identified stiffness parameter of elements can be compared with the expected values or with design values available from the drawings, or by comparing with previous results if periodic evaluations are made (say yearly or every two years), or studying variations between different elements where the values are expected to be similar. For the SHA, the objective is not to identify the exact stiffness parameters but to detect Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

6 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION variations in a relative sense or changes in the magnitude in the absolute sense or both. The theoretical concept of the GILS-EKF-UI method is presented in this paper. GILS-EKF-UI CONCEPT As discussed earlier, to identify a structure using only limited response information and without using any excitation information in the presence of several sources of uncertainty, a two-stage approach is necessary. They are as follows. Stage : Based on the response information, a substructure model can be developed that will satisfy all the requirements for the MILS-UI method proposed by Ling and Haldar [42]. It will generate information on the unknown excitation force fðtþ and the stiffness of all the elements in the substructure and the Rayleigh damping coefficients. The information on excitation force and the initial values of the state vectors of the substructure can be judiciously used to satisfy all the requirements for the EKF-WGI-based method. Stage 2: The EKF-WGI method can now be implemented to identify the whole structure, providing the conceptual basis of the GILS-EKF-UI method as discussed next. GILS-EKF-UI}Stage Structural representation using finite elements is desirable for a Kalman filter-based structural identification method since most structures of practical importance are generally represented by them. The finite element-based approach is used to develop the proposed method. Without losing any generality and for ease of discussion, two-dimensional frame structures represented by beam elements are used to present the underlying theory in this paper. The selection of the substructure by satisfying all the requirements of the MILS-UI method is the first important step in Stage. To implement the MILS-UI method, responses at all DDOFs of the substructure and the point(s) of application of the excitation force(s) must be available. Since the measured response information is limited, this will require considering one or more substructures in the system to be identified. The selection of substructure(s) will depend on the instrumentation of the structure or the locations where the responses were measured, assuming that it may be practically impossible to completely instrument a realistic large structural system. The same substructure selection criteria can be used for the seismic loading also. The following six steps can be used to define an appropriate substructure. Step : Identify the key node(s). These are the node(s) where unknown excitation force(s) is applied. For earthquake loading, all nodes in the structure can be considered to be key nodes since the inertia forces resulting from the seismic load are applied to all nodes. Step 2: Determine the total number of DDOFs at the key node(s), Ndkey. For twodimensional frames, the number of DDOFs for each key node is three: two translational DDOFs (one along the length of the element (x-axis) and the other perpendicular to the x-axis, i.e. along the y-axis) and a third DDOF representing the rotation of the node. Step : Determine the number of related elements, nerl. These elements are connected directly to the key node(s) in the finite element representation and contribute to the equilibrium equations of the substructure. Step 4: Determine the number of related nodes, nrl. These nodes are attached to the related elements in the finite element representation and contribute to the equilibrium equations of the Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

7 STC : 22 6 H. KATKHUDA AND A. HALDAR substructure. Thus, the total number of nodes in the substructure, nnsub, is the summation of the number of key node(s) and the nrl, i.e. nnsub ¼ keynodeðsþþnrl: Step : Determine the total number of elements in the substructure, nesub. It can be calculated as nesub ¼ nerl þ nerln; where nerl is defined in Step and nerln is the number of elements connecting the related nodes not included in nerl. Step 6: Determine the number of DDOFs in the related nodes, ndrl, and the total number of DDOFs for the substructure, Ndsub. It can be shown that ndrl ¼ nrl nddof; where nddof is the number of DDOFs in a related node. Accordingly, Ndsub ¼ ndrl þ Ndkey: Katkhuda and Haldar [4] discussed the substructure identification process in detail with examples in terms of the six steps just discussed. For economic reasons, i.e. to identify a structure with minimum response information, the size of the substructure should be kept to a minimum. The tasks in Stage are to identify all the elements in the substructure using the MILS-UI procedure and the unknown input excitation. The governing equation of motion for the substructure using Rayleigh-type damping can be expressed as K sub x sub ðtþþðam sub þ bk sub Þ x sub ðtþþm sub.x sub ðtþ ¼f sub ðtþ ðþ where K sub and M sub are the global stiffness and mass matrices, respectively,.x sub ðtþ; x sub ðtþ; x sub ðtþ the vectors containing acceleration, velocity, and displacement at time t; respectively, a the mass-proportional damping coefficient, b the stiffness-proportional damping coefficient, and f sub ðtþ the unknown excitation force vector(s) acting at the key node(s). M sub can be assembled from the mass matrices of all the elements in the substructure by considering their connectivity and following the standard finite element formulation as [2] M sub ¼ Xnesub M i ð2þ where M sub is a matrix of size ðndsub NdsubÞ; Ndsub is the total number of DDOFs to represent the sub-structure, nesub the total number of elements in the substructure, and M i the consistent mass matrix for the ith beam element in the substructure. For a twodimensional beam element of uniform cross section with three DDOFs at each node, it can be represented as [2] Sym: 0 22L i 4L 2 M i i ¼ %m i L i =420 ðþ L i L i L 2 i 0 22L i 4L 2 i i¼ where L i and %m i are the length and mass per unit length of the ith element, respectively. The mass matrix is generally assumed to be known [4]. K sub can be similarly assembled from the stiffness matrices of all the elements in the substructure using the direct stiffness method and Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

8 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION their connectivity as [2] K sub ¼ Xnesub where K i is the stiffness matrix for the two-dimensional ith beam of uniform cross section (constant flexural stiffness or constant EI) in the substructure and is given by 2 A i =I i 0 0 A i =I i =L 2 i 6=L i 0 2=L 2 i 6=L i 0 6=L i 4 0 6=L i 2 K i ¼ E i I i =L i ðþ A i =I i 0 0 A i =I i =L 2 i 6=L i 0 2=L 2 i 6=L i 4 0 6=L i 2 0 6=L i 4 where E i ; I i ; and A i are Young s modulus, moment of inertia, and area of the cross section of the ith beam element, respectively. Equation () can be rewritten as K i ¼ k i S i ð6þ where k i ¼ E i I i =L i and S i is a matrix of size 6 6; represented by all the elements in the square bracket in () for the ith element. Combining (4) and (6) results in K sub ¼ Xnesub k i S i ¼ k S þ k 2 S 2 þþk nesub S nesub ðþ i¼ Incorporating all the information, () can be rewritten as k S x sub ðtþþk 2 S 2 x sub ðtþþþk nesub S nesub x sub ðtþ þ bk S x sub ðtþþbk 2 S 2 x sub ðtþþþbk nesub S nesub x sub ðtþþam sub x sub ðtþ ¼ f sub ðtþ M sub.x sub ðtþ ð8þ Suppose the response of the substructure is measured for duration of h Dt at all DDOFs, where h is the number of sample points and Dt is the constant time increment. Then for a known value of h and assuming that the response information is available at all DDOFs of the substructure, i.e. at Ndsub locations, (8) can be represented in a matrix form as A ðndkeyhþlsub P Lsub ¼ F ðndkey:hþ ðþ where A is a matrix of size ðndkey hþlsub; Ndkey the total number of DDOFs for the key node(s) in the substructure as discussed earlier, P the a vector of unknown parameters to be identified in the substructure, Lsub the total number of unknown parameters in the substructure, and F the vector composed of the unknown input excitation and inertia forces. In general, the A matrix can be expressed as A ðndkeyhþlsub ¼½S x sub ðtþ S 2 x sub ðtþ S nesub x sub ðtþ S x sub ðtþ S 2 x sub ðtþ i¼ S nesub x sub ðtþ M sub x sub ðtþš K i ð4þ ð0þ Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

9 STC : 22 8 H. KATKHUDA AND A. HALDAR For two-dimensional beam elements, x sub ðtþ and x sub ðtþ vectors at the time t can be expressed as x sub ðtþ ¼½x i ; y i ; y i ;...; x nnsub ; y nnsub ; y nnsub Š T ðþ and x sub ðtþ ¼½ x i ; y i ; y i ;...; x nnsub ; y nnsub ; y nnsub Š T ð2þ where x i ; y i ; and y i are the horizontal, vertical, and rotational displacements, respectively, at the ith node, x i ; y i ; and y i the horizontal, vertical, and rotational velocities, respectively, at the ith node, and nnsub the total number of nodes in the substructure, as stated earlier. P vector in () contains the unknown system parameters and can be expressed as P ¼½k ; k 2 ;...; k nesub ; bk ; bk 2 ;...; bk nesub ; aš T ðþ where k i ¼ E i I i =L i is the unknown stiffness parameter for the ith beam element and a and b the Rayleigh damping coefficients, as discussed earlier. F matrix in () can be expressed as F ðndkeyhþ ¼ f subðndkey:hþ M sub.xðtþ subðndkeyhþ ð4þ where f is the unknown excitation force vector. In general, the structure can be excited at any or all DDOFs including at the base. It can be expressed at time t as fðtþ ¼½f ðtþ f 2 ðtþ f Ndkey ðtþ f Ndkey ðtþš T ðþ.x sub ðtþ vector in (4) contains the acceleration responses at all DDOFs of the substructure at the time t and can be expressed as.x sub ðtþ ¼½.x i ;.y i ; ẏ i;...;.x nnsub ;.y nnsub ; ẏ nnsubš T ð6þ where.x i ;.y i ; and ẏ i are the horizontal, vertical, and rotational accelerations, respectively, at the ith node, and nnsub the total number of nodes in the substructure, as stated earlier. For mathematical convenience, () can also be expressed as XLsub s¼ A rs P s ¼ F r ; r ¼ ; 2;...; h N Ndkey ðþ Suppose #P s is the predictor of the sth parameter P s of the substructure. Considering all the Lsub parameters that need to be identified for the substructure, the total error function Er in the estimation of the parameters can be expressed as Er ¼ X hndkey r¼! 2 F r XLsub A rs #P s ; r ¼ ; 2;...; h Ndkey ð8þ s¼ The least-squares technique is used to evaluate all the unknown parameters of the substructure. To minimize the total error, (8) can be differentiated with respect to each one of the #P q parameters #P q ¼ X hndkey r¼ F r XLsub A rs P s!a rq ¼ 0; q ¼ ; 2;...; Lsub ðþ s¼ Equation () gives Lsub simultaneous equations. The solution of the simultaneous equations will give all Lsub unknowns parameters to be estimated in the substructure. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

10 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION It is relatively simple to solve for the system parameters vector P in () provided that the force vector F and A are known. However, as mentioned earlier, the input excitation fðtþ is considered to be unknown; thus, the force vector F in () is partially known and the iteration process cannot be initiated. To start the iteration process, the input excitation force fðtþ is assumed to be zero for few time points [4]. Later, Katkhuda et al. [, 4] observed that if fðtþ was assumed to be zero for all h time points, the iterative process gave better results. The flow chart for Stage is shown in Figure. At the completion of Stage, the time history of the unknown excitation force, the Rayleigh-damping coefficients, and the stiffness parameters of all the elements in the substructure will be available. The information on damping will be applicable to the whole structure. The identified stiffness parameters for the substructure can be judiciously used to develop the initial state vector of the stiffness parameters for the whole structure as discussed in detail in Stage 2. No Start Select the Substructure(s) needed Assume Initial Input excitation force f (t) = 0 for all h Time Points Develop [A] [P] = [F] equation for the selected Substructure(s) based on equation (0) Solve for P LSUB Obtain f (t) at all Time Points h Update P LSUB and f (t) Check Convergence i + i f f ε Yes End Input excitation force f (t), damping coefficient α and β, and the stiffnesses of the elements in the substructure are determined Figure. Flow chart for GILS-EKF-UI Stage. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

11 STC : 22 0 H. KATKHUDA AND A. HALDAR GILS-EKF-UI}Stage 2 In Stage, the mathematical model used is considered to be perfect and the measured response information is considered to be noise free. As mentioned earlier, Maybeck [] commented that deterministic system and control theories were not sufficient for identifying the system using responses obtained by measurement devices. The imperfect nature of mathematical modeling and noise in the measured responses need to be incorporated in Stage 2, and the EKF method will be ideal to address them. In order to apply the EKF method, the state vector can be defined as 2 2 Z ðtþ XðtÞ ZðtÞ ¼6 4 Z 2 ðtþ ¼ 6 XðtÞ 4 ð20þ Z ðtþ *K where ZðtÞ is the state vector at time t; XðtÞ and XðtÞ the displacement and velocity vectors, respectively, at time t for the whole structure, and *K the a vector containing the element stiffness parameters of the whole structure that need to be identified. For two-dimensional beam elements, these vectors can be expressed as 2 2 x ðtþ x ðtþ y ðtþ y ðtþ y ðtþ y ðtþ x 2 ðtþ x 2 ðtþ 2 k y 2 ðtþ y 2 ðtþ k 2 XðtÞ ¼ y 2 ðtþ ; XðtÞ ¼ ; *K ¼ y 2 ðtþ. ð2þ k ne x N ðtþ x N ðtþ 6 y N ðtþ y N ðtþ y N ðtþ y N ðtþ where N and ne are the total number of DDOFs and elements in the whole structure, respectively, and k i ¼ E i I i =L i is the unknown stiffness parameter for the ith beam element. It is assumed that the stiffness will not change with time during the identification process. The equation of motion can be expressed in a state equation as Z ðtþ XðtÞ XðtÞ ZðtÞ ¼6 4 Z 2 ðtþ ¼ 6 4 ẊðtÞ ¼ 6 M ðkxðtþþðamþbkþ XðtÞ fðtþþ 4 ð22þ Z ðtþ where K and M are the global stiffness and mass matrixes of the whole structure, respectively, and can be evaluated using equations similar to (2) and (4), except nesub needs to be replaced by Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

12 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION ne to carry out the summation for the whole structure, and fðtþ is the input excitation force vector identified in Stage. In general, (22) can be mathematically expressed as dz dt ¼ ZðtÞ ¼f ½ZðtÞ; tš ð2þ The initial state vector Z 0 is assumed to be Gaussian with a mean vector of #Z 0 and a constant error covariance matrix of D: It is generally denoted as Z 0 Nð #Z 0 ; DÞ: Suppose the response of the structure is measured at time t k and t k ¼ kdt; where Dt is the time interval between the measurements. Then the observational vector Y tk at time t k can be expressed as Y tk ¼ HZðt k ÞþV tk ð24þ where Y tk is the observational vector of size ðb Þ; B the total number of displacement and velocity observations (the information of acceleration is not required for the second stage), Zðt k Þ the state vector of size ð2n þ LÞ at time t k ; L the total number of unknown stiffness parameters, H the a matrix of size ½B ð2n þ LÞŠ containing information of measured responses, and V tk the observational noise vector of size ðb Þ; assumed to be Gaussian white noise with zero mean and a covariance of R tk : It is generally denoted as V tk Nð0; R tk Þ: Stage 2 can be carried out in the following steps. Step : Define the initial state vector #Z 0 ðt 0 =t 0 Þ and its error covariance Dðt 0 =t 0 Þ: The initial state vector can be expressed as 2 2 Z ðt 0 =t 0 Þ Xðt 0 =t 0 Þ #Z 0 ðtþ ¼6 4 Z 2 ðt 0 =t 0 Þ ¼ 6 Xðt 0 =t 0 Þ 4 ð2þ Z ðt 0 =t 0 Þ *Kðt 0 =t 0 Þ where Xðt 0 =t 0 Þ and Xðt 0 =t 0 Þ are the displacement and velocity vectors, respectively. They can be obtained from (2) where Xðt 0 =t 0 Þ¼XðtÞ and Xðt 0 =t 0 Þ¼ XðtÞ: Only acceleration responses will be measured at a few DDOFs to implement the procedure. The acceleration time histories will be successively integrated to obtain the required velocity and displacement time histories []. The initial parts of acceleration, velocity, and displacement time histories, say for the duration between 0.0 and 0.0 s, were not used for the identification purpose to avoid the effect of initial boundary conditions and the integration error. The initial value of the stiffness parameters for all the elements in the structure *Kðt 0 =t 0 Þ can be assumed to be 2 2 k ðt 0 =t 0 Þ k i k 2 ðt 0 =t 0 Þ k i *Kðt 0 =t 0 Þ¼ k ðt 0 =t 0 Þ ¼ k i ð26þ k ne ðt 0 =t 0 Þ Assuming the substructure in Stage has only two elements, one beam and another column, and their identified stiffness parameters are k and k 2 ; then k i ðt 0 =t 0 Þ in (26) can be assumed as k or k i Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

13 STC : 22 2 H. KATKHUDA AND A. HALDAR k 2 depending upon whether the element under consideration is a beam or a column. Conceptually, any initial values of k i s can be assumed. However, if the initial assumed values are far from the actual values, it may create a convergence problem or reduce the efficiency of the algorithm. Stiffness parameters of all the elements in the similar category in a real structure are expected to be similar; thus, this assumption is quite reasonable. The initial error covariance matrix Dðt 0 =t 0 Þ contains information on the errors in the velocity and displacement responses and in the initial estimate of the stiffness parameters of the elements. It is generally assumed to be a diagonal matrix and can be expressed as [, ] " Dðt 0 =t 0 Þ¼ D # xðt 0 =t 0 Þ 0 ð2þ 0 D k ðt 0 =t 0 Þ where D x ðt 0 =t 0 Þ is a matrix of size ð2n 2NÞ and N is the total number of DDOFs in the whole structure. This represents the initial errors in the velocity and displacement responses, and is assumed to have a value of.0 in the diagonals. D k ðt 0 =t 0 Þ is a diagonal matrix of size ðl LÞ and L is the total number of the unknown stiffness parameters to be identified, as mentioned earlier. It contains the initial covariance matrix of *K: Hoshiya and Saito [] and Jazwinski [4] pointed out that the diagonals should be large positive numbers to accelerate the convergence of the local iteration process, and 000 can be used for this purpose. The same value is used in this study. Step 2 (Prediction phase): In the context of EKF, the predicted state #Zðt kþ =t k Þ and its error covariance Dðt kþ =t k Þ are evaluated by linearizing the nonlinear dynamic equation as Z tkþ #Zðt kþ =t k Þ¼ #Zðt k =t k Þþ f ½ #Zðt=t k Þ; tš dt ð28þ t k and Dðt kþ =t k Þ¼U½t kþ ; t k ; #Zðt k =t k ÞŠ Dðt k =t k ÞU T ½t kþ ; t k ; #Zðt k =t k ÞŠ ð2þ where U½t kþ ; t k ; #Xðt k =t k ÞŠ is the state transfer matrix from time t k to t kþ and can be written in an approximate form as U½t kþ ; t k ; #Zðt k =t k ÞŠ ¼ I þ Dt F½t k ; #Zðt k =t k ÞŠ ð0þ In which, I is a unit matrix and F½t k ; #Zðt k =t k ÞŠ ½Zðt kþ; t k Š j Zðt k Þ¼ Zðt # k =t k Þ where Z j is the jth component of the vector Zðt k Þ: Step (Updating phase): Since observations are available at k þ ; the state vector and the error covariance matrix can be updated as #Zðt kþ =t kþ Þ¼ #Zðt kþ =t k ÞþK½t kþ ; #Zðt kþ =t k ÞŠ fyðt kþ Þ H½#Zðt kþ =t k Þ; t kþ Šg ð2þ Dðt kþ =t kþ Þ¼fI K½t kþ ; #Zðt kþ =t k ÞŠ M½t kþ ; #Zðt kþ =t k ÞŠg Dðt kþ =t k Þ fi K½t kþ ; #Zðt kþ =t k ÞŠ M½t kþ ; #Zðt kþ =t k ÞŠg T þ K½t kþ ; #Zðt kþ =t k ÞŠ Rðt kþ ÞK T ½t kþ ; #Zðt kþ =t k ÞŠ ðþ Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

14 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION where K½t kþ ; #Zðt kþ =t k ÞŠ is the Kalman gain matrix. It can be evaluated from K½t kþ ; #Zðt kþ =t k ÞŠ ¼ Dðt kþ =t k ÞM T ½t kþ ; #Zðt kþ =t k ÞŠ fm½t kþ ; #Zðt kþ =t k ÞŠ Dðt kþ =t k ÞM T ½t kþ ; #Zðt kþ =t k ÞŠ þ Rðt kþ Þg ð4þ where M½t k ; #Zðt k =t k ÞŠ t k ; t k Þ j Step 4: By taking the next time increment, i.e. k ¼ k þ and using (28), (2), (2), and (), the system parameters are updated. This procedure will continue until all the time points are used, i.e. k ¼ h; where h represents the total number of discrete time points of the measurements. The iteration process covering all the time points is generally defined as the local iteration. When the local iteration procedure is completed, Hoshiya and Saito [] suggested incorporating a weighted global iterative procedure with an objective function into the local iteration to obtain the stable and convergent estimation of the parameters to be identified. To start the first global iteration, the initial values of #Z ðþ ðt h =t h Þ and D ðþ ðt h =t h Þ need to be assumed, or the information from Step 4 stated above can be used to increase the efficiency, where superscript () represents the first global iteration, and they can be expressed as 2 Z ðþ ðt 2 0=t 0 Þ X ðþ ðt 0 =t 0 Þ #Z ðþ ðt h =t h Þ¼6Z ðþ 2 ðt 0=t 0 Þ 4 ¼ 6 X ðþ ðt 0 =t 0 Þ 4 ð6þ Z ðþ ðt h=t h Þ *K ðþ ðt h =t h Þ and " D ðþ ðt h =t h Þ¼ D # xðt 0 =t 0 Þ 0 ðþ 0 D k ðt h =t h Þ In the second global iteration, a weight factor ðwþ is introduced to the error covariance matrix to accelerate the local EKF iteration. Hoshiya and Saito [] observed that to obtain better and stable convergence, the value of w should be a large positive number. Hoshiya and Saito [], Koh et al. [46, 4], Oreta et al. [48], Oreta and Tanabe [4] assumed w to be 00 and Hoshiya and Sutoh [4] assumed it to be in the range of in some applications. In this study, w of value 00 produced satisfactory results. To start the second global iteration, the initial values of the state vector #Z ð2þ ðt 0 =t 0 Þ and the error covariance matrix D ð2þ ðt 0 =t 0 Þ can be shown to be 2 #Z ð2þ ðt 2 0=t 0 Þ X ð2þ 2 ðt 0 =t 0 Þ X ðþ ðt 0 =t 0 Þ #Z ð2þ ðt 0 =t 0 Þ¼ #Z ðþ ðt h =t h Þ¼6 #Z ð2þ 2 ðt 0=t 0 Þ 4 ¼ 6 X ð2þ ðt 0 =t 0 Þ 4 ¼ 6 X ðþ ðt 0 =t 0 Þ 4 ð8þ #Z ð2þ ðt 0=t 0 Þ *K ð2þ ðt 0 =t 0 Þ *K ðþ ðt h =t h Þ and " # D x ðt 0 =t 0 Þ 0 D ð2þ ðt 0 =t 0 Þ¼ 0 wd ðþ k ðt h=t h Þ ðþ Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

15 STC : 22 4 H. KATKHUDA AND A. HALDAR where w is the weight factor used to accelerate the local iteration, as discussed earlier and D ðþ k ðt h =t h Þ the error covariance matrix corresponding to the parameter *K at time t h in the last global iteration. With this information, the prediction and updating phases of the local iteration are carried out for all the time points, producing the state vector and the error covariance matrix for the second global iteration. The information can then be used to initiate the third global iteration. The global iterations are repeated until a predetermined convergence criterion is satisfied, i.e. j #Z i ðt h =t h Þ #Z i ðt h =t h Þj4e; where i represents the iteration number and e is the tolerance level to be used in the numerical evaluation of the stiffness parameters. Since the stiffness parameters are of the order of 00 in this study, e is considered to be 0.. Figure 2 shows the flow chart of Stage Start i = Start with initial ˆ (i) Z 0 ( t 0 /t 0 ) and D (t0 /t0 ) using Eq. (2 and 2) Prediction Phase Predict ˆ ( i) ( i) Z ( /t ) and D ( /t ) using Eq. (28 and 2) t k + k t k + Updating Phase Calculate Kalman gain matrix K [ t ˆ k + ; Z( tk + /tk )], ˆ ( i) Z ( /t ), No ( i) and D ( /t ) using Eq. (2, and 4) ˆ ( i) t k + k+ k = Ye s k = h Estimate ˆ ( i) (i) Z ( th /th ) and D (th /th ) k k = k +; is updated k = h? Check Convergence of identified parameters i = i + (i) No Set Z ( t0 /t0 ) and D (t0 /t0 ) by Eq. (6 and ) k = t k+ k+ Ye s 4 End Figure 2. Flow chart for GILS-EKF-UI Stage 2. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

16 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION SUMMARY OF THE PROPOSED GILS-EKF-UI METHOD All the essential components of the proposed GILS-EKF-UI algorithm are shown in Figure. The components are summarized below.. Considering the available response information, select the most appropriate substructure(s) by a. Identifying the key node. b. Determining the total number of DDOFs for the key node(s), Ndkey. c. Determining the related elements, nerl. d. Determining the related nodes, nrl, and the total number of nodes in the substructure, nnsub. e. Determining the elements connecting between the related nodes, nerln, and the total number of elements in the substructure, nesub. f. Determining the number of DDOFs in the related nodes, ndrl, and the total number of DDOFs for substructure, Nsub. 2. Develop an equation of the form A ðndkeyhþlsub P Lsub ¼ F ðndkeyhþ ; as in () for the substructure. Using the MILS-UI procedure, identify the stiffness of all the elements in the substructure, the two Rayleigh damping coefficients, and the unknown excitation force(s). Start Select the most appropriate Sub-structure(s) Develop [A] [P] = [F] equation for the selected Sub-structure(s) based on Equation () Apply the Generalized MILS-UI procedure to identify the input excitation force, damping coefficients and the stiffnesses of the elements in the substructure Develop the initial conditions Z ˆ () ( t 0 /t 0 ) and () D (t 0 /t 0) and complete the local iteration using the EKF procedure Apply the weight factor (w) until the stiffnesses of the all elements in the whole structure are identified with predetermined tolerance level or the objective function reaches a minimum value End Figure. Flow chart of the GILS-EKF-UI method. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

17 STC : 22 6 H. KATKHUDA AND A. HALDAR Start the first global iteration by developing the initial state vector #Z 0 ðt 0 =t 0 Þ and its error covariance matrix Dðt 0 =t 0 Þ using (2) (2) using the information from Stage. Complete the local iterations using the EKF procedure. 4. Continue the weighted global iterations by applying a weight factor ðwþ until the stiffnesses of all the elements of the structure are identified with a predetermined tolerance level (e). A computer program was developed to implement the procedure. The proposed GILS-EKF-UI procedure was verified using computer-generated theoretical response information and using experimentally generated response information [6]; however, they cannot be discussed here due to lack of space. The method appears to be very efficient and accurate in identifying defects in structures in the presence of several sources of uncertainty, using limited response information and without using excitation information. CONCLUSIONS A novel structural health assessment method is proposed to detect defects at the element level using only minimum response information considering imperfect mathematical model representing the structure, various sources of uncertainty in the mathematical model, and uncertainty in the measured response information. It is a linear time domain finite element-based system identification procedure capable of detecting defects at the local element level using only limited response measurements and without using any information on the exciting dynamic forces. The technique is a combination of iterative least-squares technique with unknown input excitation (MILS-UI) proposed earlier by the research team at the University of Arizona and the extended Kalman filter with a weighted global iteration (EKF-WGI) procedures. The authors denote the new method as GILS-EKF-UI. To implement the concept, a two-stage substructure approach is used. In the first phase, a substructure is selected that satisfies all the requirements for the MILS-UI procedure. This provides information on the initial state vector and the input excitation required to implement any EKF-based procedure. In the second stage, the EKF-WGI is applied to identify the whole structure. This way the whole structure, defect-free or defective, can be identified with limited response measurements and without using excitation information. The theoretical concept behind the method is presented in this paper. The method is then verified in the companion paper using experimentally measured and analytically generated limited response information, but cannot be presented here due to lack of space. The feasibility and application potential of this novel concept are conclusively established in this study. ACKNOWLEDGEMENTS This paper is based on work partly supported by University of Arizona Foundation. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsor. REFERENCES. Doebling S, Farrar C, Prime M, Shevitz D. Damage identification and health monitoring of structural and mechanical systems from changes in their variation characteristics: a literature review. Report No. LA-00-MS, Los Alamos National Laboratory, 6. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22

18 STC : 22 NOVEL HEALTH ASSESSMENT TECHNIQUE WITH MINIMUM INFORMATION Housner G, Bergman L, Caughey T, Chassiakos A, Claus R, Masri S, Skelton R, Soong T, Spencer B, Yao J. Structural control: past, present and future. Journal of Engineering Mechanics (ASCE) ; 2():8.. Maybeck P. Stochastic Models; Estimation; and Control, chapter. Academic Press: London, U.K.,. 4. Alampalli S, Fu G. Signal versus noise in damage detection by experimental modal analysis. Journal of Structural Engineering (ASCE) ; 2(2): Barroso L, Rodriguez R. Damage detection utilizing the damage index method to a benchmark structure. Journal of Engineering Mechanics (ASCE) 2004; 0(2): Beck J, Yuen K. Model selection using response measurements: Bayesian probabilistic approach. Journal of Engineering Mechanics (ASCE) 2004; 0(2): Lam HF, Katafygiotis S, Mickleborough NC. Application of a statistical model updating approach on phase I of the IASC-ASCE structural health monitoring benchmark study. Journal of Engineering Mechanics (ASCE) 2004; 0(): Law SS, Shi YZ. Zhang LM. Structural damage detection from incomplete and noisy modal test data. Journal of Engineering Mechanics (ASCE) 8; 24(): Shi ZY, Law SS, Zhang LM. Improved damage quantification from elemental modal strain energy change. Journal of Engineering Mechanics (ASCE) 2002; 28(): Yuen KV, Au SK, Beck J. Two-stage structural health monitoring approach for phase I benchmark studies. Journal of Engineering Mechanics (ASCE) 2004; 0():6.. Vestroni F, Capecchi D. Damage detection in beam structures based on frequency measurements. Journal of Engineering Mechanics (ASCE) 2000; 26(): Ibanez P. Identification of dynamic parameters of linear and nonlinear structural models from experimental data. Nuclear Engineering Design 2; McCann D, Jones NP, Ellis JH. Toward consideration of the value of information in structural performance assessment. Structural Engineers World Congress, 8; Paper No. T26-6 (CD-ROM). 4. Wang D, Haldar A. An element level SI with unknown input information. Journal of the Engineering Mechanics Division (ASCE) 4; 20(): 6.. Wang D, Haldar A. System identification with limited observations and without input. Journal of Engineering Mechanics (ASCE); 2(): Lin JW, Betti R, Smyth AW, Longman RW. On-line identification of nonlinear hysteretic structural systems using a variable trace approach. Earthquake Engineering & Structural Dynamics 200; 0():2 0.. Loh C, Lin CY, Huang CC. Time domain identification of frames under earthquake loadings. Journal of Engineering Mechanics (ASCE) 2000; 26(): Yang JN, Lin S. Identification of parametric variations of structures based on least squares estimation and adaptive tracking technique. Journal of Engineering Mechanics (ASCE) 200; : Hoshiya M, Saito E. Structural identification by extended Kalman filter. Journal of Engineering Mechanics (ASCE) 84; 0(2): Sato T, Honda R, Sakanoue T. Application of adaptive Kalman filter to identify a five story frame structure using NCREE experimental data. Proceedings of ICOSSAR, 200 (CD-ROM). 2. Yang JN, Lin SL, Huang HW, Zhou L. An adaptive extended Kalman filter for structural damage identification. Structural Control and Health Monitoring 200 ( 22. Yang JN, Pan S, Huang H. An adaptive extended Kalman filter for structural damage identifications II: unknown inputs. Structural Control and Health Monitoring Published online in Wiley Interscience ( wiley.com). 2. Sato T, Qi K. Adaptive H filter: its application to structural identification. Journal of Engineering Mechanics (ASCE) 8; 24(): Yoshida I. Damage detection using Monte Carlo filter based on non-gaussian noise. Proceedings of ICOSSAR 200 (CD-ROM). 2. Hoshiya M, Maruyama O. Identification of running load and beam system. Journal of Engineering Mechanics (ASCE) 8; (4): Safak E. Adaptive modelling, identification and control of dynamic structural systems I: theory. Journal of Engineering Mechanics (ASCE) 8; Bedewi NE. The mathematical foundation of the auto and cross random decrement techniques and the development of a system identification technique for the detection of structural deterioration. Ph.D. Thesis, The University of Maryland, Hac A. Spanos PD. Time domain method for parameter system identification. Journal of Vibration and Acoustics 0; 2: Kung DN, Yang JCS, Bedewi NE, Tsai WH. Time domain system identification technique based on impulsive loading for damage detection. Proceedings of 8th International Symposium on Offshore Mechanics and Arctic Engineering, 8; Smith KE. An evaluation of a least-square identification technique based on free-vibration decay responses for damage detection. Proceedings of Conference on MFPG 42, NBS, MD, 8. Copyright # 200 John Wiley & Sons, Ltd. DOI: 0.002/stc stc 22