Abstract
A parameter identification method and a high gain observer are proposed in order to identify the model and to recover the state of a seismically excited shear building using acceleration responses of the ground and instrumented floors levels, as well as the responses at noninstrumented floors, which are reconstructed by means of cubic spline shape functions. The identification method can be implemented online or offline and uses Linear Integral Filters, whose bandwidth must enclose the spectrum of a seismically excited building. On the other hand, the proposed state observer estimates the displacements and velocities of all the structure floors using the model estimated by the identification method. The observer allows obtaining a fast response and reducing the state estimation error, while depending on a single gain. The performance of the parameter and state estimators is verified through experiments carried out on a five-story small scale building.
1. Introduction
Acceleration responses of the building floor, obtained when it is excited through earthquakes or wind, are employed for parameter and state estimation of the structure or for monitoring its health. Because of cost, most of the buildings are instrumented only at a few stories. For this reason, the estimation and health monitoring of a structure must be carried out using as data only the responses of the instrumented stories.
Parameter and state estimation of civil structures using recorded responses has attracted the attention of a large number of researchers around the world during the last four decades. Some relevant references for not fully instrumented structures [1–15] are revised here. Amini and Hedayati [1] propose a Sparse Component Analysis approach for modal identification of structures, where the number of sensors is smaller than the number of active modes. References [2–5] employ the extended Kalman filter in order to estimate the parameters of a structure, as well as its displacements and velocities at noninstrumented floors. This filter is obtained by linearizing a nonlinear state equation that considers the building parameters as states. Nevertheless, some poles of the linearized model may lie on the imaginary axis and, as a consequence, some parameter and/or states may be unbounded. On the other hand, Zhou et al. [6] recursively estimate the stiffness and damping ratios of a structure using the first two derivatives of Log-Likelihood Measure and the knowledge of the building mass. Mukhopadhyay et al. [7] develop mode shape normalization and expansion approaches that utilize the topology of the structural matrices for estimating the mass and stiffness parameters of a building under base excitation; authors obtain the global identifiability requirements for their methodology and show that, for estimating the structural parameters, the number of instrumented floors should satisfy and if the floor masses at sensor locations are known and if the total mass of the structure is known, respectively, where is the number of floors of the building. Yuan et al. [8] estimate the mass and stiffness matrices of shear buildings using a method depending on the first two orders of modal data. References [9–11] present OKID identification techniques, which determine the Markov parameters of a Kalman observer in order to identity a building model. These techniques identify a discrete time model of the structure, but if the sampling frequency of responses is too high in relation to the dominate frequencies of the structure, then the poles of the estimated model lie close to the unit circle in the complex domain, which may produce parameter estimates statistically ill-defined [16]. Kaya el al. [12] identify a structure using the Transfer Matrix formulation of the response, where the histories at noninstrumented floors are offline reconstructed using the Mode Shape Based Estimation (MSBE) method, which assume that modal shapes can be approximated as a linear combination of the mode shapes of a shear beam and a bending beam. It is worth mentioning that using MSBE requires solving two partial differential equations and the knowledge of the modal acceleration of the instrumented floors. Hegde and Sinha [13] estimate the modal parameters of a seismically excited, torsionally coupled building using limited number of responses; modal parameters are extracted using an offline estimation methodology based on the principles of the Natural Excitation Technique and the Eigen Realization Algorithm; authors use a cubic shape function for estimating the responses at noninstrumented floors. On the other hand, [14, 15] carried out structural identification using ambient vibration measurements and offline algorithms. Huang [14] proposes a procedure for identifying the dynamics characteristics of a shear building using the multivariate ARV model, whereas Chakraverty [15] estimates mass and stiffness parameters from modal test data and the Holzer criteria.
This manuscript presents an identification technique to identify the parameters of the model of a seismically excited shear building using a limited number of responses and a high gain state observer that employs the identified parameter in order to estimate the displacements and velocities of both instrumented and noninstrumented floors. The parameter and state estimators rely on the acceleration measurements of ground and some instrumented floors. The acceleration at noninstrumented floors is reconstructed by means of cubic shape spline functions that use the available measurements. The state observer gain is easily designed and depends on a positive parameter that guarantees the stability and a fast response of the observer. On the other hand, the parameter estimator uses a parameterization with the following features: assuming that building has Rayleigh damping; containing a vector whose entries are the stiffness/mass ratios of the building, which are estimated through a Least Squares algorithm; and employing Linear Integral Filters (LIF) that attenuate low and high frequency noise of the measured and reconstructed responses. The LIF were previously employed in Garrido and Concha [17] for parameter estimation of fully instrumented structures. In this manuscript, the application of these filters is extended to estimate the model of buildings instrumented at only few floor levels. Moreover, by considering Rayleigh damping for the structure, the number of parameters of the proposed parameterization is the half of the one corresponding to the parameterization in [17]. In comparison with techniques in [7–15] described above, the proposed parameter estimator allows attenuating low and high frequency noise of both measured and reconstructed acceleration and can be implemented online at a low computational effort.
It is worth mentioning that the cubic spline functions, employed by the proposed parameter and state estimators, are designed following the ideas in [18, 19]. Moreover, the design of these functions does not require the knowledge of the modal shapes of the structure, and they allow recursive estimation of the responses at noninstrumented floors. Techniques in [20–26] are also able to estimate the unmeasured floor level responses, but unlike the cubic spline functions they are implemented offline and require an estimated building model, which may be unavailable.
The paper has the following structure. Sections 2 and 3 present the shear building model and its parameterization, respectively. The cubic spline functions used for estimating the floor responses at noninstrumented floors are described in Section 4. Section 5 shows the methodology for estimating the building parameters. The proposed high gain observer is described in Section 6. Experiment results using the parameter and state estimators are shown in Section 7. Finally, Section 8 includes the conclusions of this paper.
2. Mathematical Model of a Shear Building
The dynamics of a shear building with stories subjected to earthquake or base motion is described through the following mathematical model [27]:where variables , , and are, respectively, the displacement, velocity, and acceleration of the th floor, which are measured with respect to the basement. Signal represents the absolute acceleration at the th floor, and is the ground acceleration produced by earthquake. Moreover, and are the mass and stiffness matrices, respectively, which are defined as The entry is the column lateral stiffness between the th and th floors.
The building has Rayleigh damping that is represented through the following matrix :Parameters and are constant, and they are computed as [27]where and , , are the damping ratio and natural frequency of the th structural mode, respectively.
Remark 1. Computing the constants and in (6) requires knowledge of the parameters , , , and that corresponds to the th and th structural modes. Natural frequencies and can be extracted by means of the Fourier spectra of the building responses produced by ambient or force excitations. On the other hand, the parameters and can be obtained with any of the following techniques: (i)Half-power bandwidth method [28], which estimates the damping ratios using the peaks of the response Fourier spectra(ii)Logarithmic Decrement method, which computes the damping ratios in the time domain by means of the ratio between the amplitudes of any two closely adjacent peaks of acceleration responses [29](iii)Enhanced Frequency Domain Decomposition method that estimates the modal damping from the Singular Value plots of recorded acceleration [30](iv)Using the recommended damping values shown in Table from [27], which depend on the type and condition of the structure.
The state space model corresponding to the differential equation (1) is given bywhere and are the identity and zero matrices with size , respectively.
The output equation of the state space model depends on the absolute acceleration of the instrumented floors. Let be the number of instrumented floors, and let be the th instrumented floor numbered from bottom to top of the building. Then, the output equation is given bywhere is the localization matrix of the accelerometers, whose entries are defined as
3. Model Parameterization
Assume first that all the stories are instrumented; it means that ; then substituting (5) into (1) yieldsEquation (12) can be rewritten asDefine the following equalities:where is the time derivative of andUsing equalities in (14) leads to the following expression:
In order to use only acceleration measurements and to attenuate low and high frequency measurement noise, (16) is first derived three times with respect to time, and, subsequently, the resulting expression is integrated five times over finite time intervals, thus obtainingThe superindex denotes the th time derivative of the corresponding signal; and is a Linear Integral Filter (LIF). The general definition a LIF is given bywhere is the number of integrations, and is the time integration window length defined as where , and is the sampling period of .
The Laplace transform of in (18) is given by [17]Terms and determine the bandwidth of the th-order low-pass filter in rad/s and Hz, respectively.
Equation (17) is equivalent to the following parameterization, which will be employed for parameter identification purposes:
It is important to mention that variables and can be written in the Laplace domain aswhere
Remark 2. The filters , , are band pass filters, and they are designed to encompass the frequency band of the seismically excited building and to attenuate low and high frequency measurement noise.
4. Estimation of the Responses at Noninstrumented Floor Levels
In practice, most of the buildings are instrumented in only some floors; it means that not all signals , which appear in variables , , and of parameterization (22), are available. In order to implement , and , the acceleration types at noninstrumented floors are reconstructed by means of cubic spline shape functions, which are described in this section.
Consider a building with height and stories that is instrumented at its basement and at floors, as shown in Figure 1. Moreover, let and be the height and acceleration response at the basement, respectively. Similarly, terms and , , are the height and absolute acceleration at the th instrumented floor, respectively, where .
(a) Building instrumented at the top floor
(b) Building noninstrumented at the top floor
Let be the absolute acceleration of the building at height . Using (3) yields and . In addition, let be the height of the th noninstrumented floor that is located within the subinterval delimited by two instrumented floors with heights and , where . The response at this noninstrumented floor is given by , or, equivalently, according to definition (3). An estimate of is computed through the following cubic spline shape function:where , , , and are the coefficients of the th cubic polynomial, which are computed at every sampling instant from continuity of the spline function, from responses , , at the instrumented floors, and from the boundary conditions of the absolute acceleration of the building. These conditions assume that the building behaves as a cantilever, and they are shown in Table 1, where superscripts , , and indicate the first, second, and third derivative with respect to the spatial variable . Appendices A and B present the cubic spline function obtained if the acceleration response at the top floor is available or not, respectively.
Once absolute acceleration of the th noninstrumented floor has been obtained, it is possible to estimate its relative acceleration through the expression . Then, the unavailable responses and in variables , and are replaced by their estimates and , respectively. From now on, variables constructed with recorded and reconstructed responses are denoted as , , and , which are given bywhere terms , , and depend on the error of the reconstructed responses and on the noise of the recorded responses.
5. Parameter Estimation of the Building
Substituting signals , , and given in (28) into (22)–(24) yieldswhere variables and have the same structure as and , respectively. However, and contain the terms , , and instead of , , and . On the other hand, vector depends on , , and . The Laplace transform of is given by
Employing (20) and (30) allows obtaining the next Laplace transform of where , , were previously defined in (26).
Expression (29) can be rewritten aswhere , , are the sampling instants of signals and . Omitting in (32) leads to
In order to estimate the parameter vector in (33), the Least Squares (LS) algorithm is employed, which is defined as [31]where is the covariance matrix; is the number of samples of and . Note that vector can be computed only if matrix exists.
On the other hand, can also be recursively identified since the responses at noninstrumented floor levels can be computed at every sampling period . The recursive version of the LS, denoted as RLS, is given bywhere is the forgetting factor such that ; moreover, is the output estimation error.
Proposition 3. Suppose that input signal is persistently exciting at least of order , where is the number of parameters of to be estimated, then vector norm of the parameter estimation error , defined as , is bounded. Moreover, the smaller the perturbation term in (33), the smaller the value of .
Proof. It is given in books [31, 32].
5.1. Estimation of Modal Parameters
Once that vector has been estimated with the offline or the online estimation methodology, it is possible to identify the natural frequencies and modal damping factors of the building. To this end, the following matrix , which is an estimate of matrix given in (8), is constructed:where parameters , , are the entries of vector . Note that (38) is deduced from (5).
The eigenvalues of the matrix in (36) are given bywhere and . Moreover, and , , are, respectively, the estimates of the natural frequency and damping factor corresponding to the th mode of building model (1).
From (39), the following equations for computing the parameters and are obtained:
5.2. Estimation of Matrices , , and
Assume that mass of the building is known; then matrix and the entries of matrices and are given byA similar procedure can be carried out if another floor mass is known instead of .
6. High Gain State Observer
The proposed high gain state observer employs the building model estimated by the LS method. This observer estimates the complete state of a building instrumented at only few floor levels, and it is given bywhere is an estimate of , is the observer gain matrix, variable is established in (28), term is the absolute acceleration estimated by the observer, matrix is presented in (36), and matrix , which is an estimate of in (10), is defined aswith and shown in (37) and (38), respectively.
Define the state estimation error as follows:Then, the state estimation error dynamics is given bywith
Proposition 4. Let be the gain of the state observer defined as(1)If satisfies where is presented in (40), and denotes the smallest value of the set , then, one has the following:(a)The eigenvalues of matrix in (48) are given by where and and are the natural frequency and damping factor of the th mode of the estimated building model, which are computed using (40).(b)Increasing the value of gain in (51)-(52) allows the real part of the eigenvalues of matrix to be more negative.(c)Norm of the estimation error is bounded, and when it satisfies where is a matrix, whose columns are the eigenvectors of matrix .(2)If , then the eigenvalues of approach and , .
Proof. See Appendix C.
6.1. Attenuation of the State Estimation Error
The norm of the state estimation error can be reduced as follows:(i)Increase in order to decrease and to obtain a fast response for the observer. Based on our experience, good results in the state estimation are obtained using a gain between and , where was defined in (52).(ii)Reduce and , which allows decreasing . According to Proposition 3, the smaller the term in (33) is, the smaller the parametric error norms and are. Signal is filtered through the filters , , and shown in (31). They are designed to include the bandwidth of the building responses and to filter measurement noise, which in turn attenuates the term and norms and .(iii)Instrument the building at regular intervals over its height, which permits reducing [19, 33]. These references also show that increasing the total number of instrumented floors decreases the value of .(iv)Attenuate the measurement noise corrupting the state observer. To achieve this, vector in (28) is replaced by the following filtered vector in variable of the state observer (42): where is the convolution operator, and is a fourth-order low-pass Butterworth filter, whose cut-off frequency is appropriately chosen.
7. Experimental Results
Figure 2 depicts a five-story small scale building with dimensions cm, which is used during the experiments. The structure is mounted over a shake table with Parker linear motors 406T03LXR, and it is built with aluminum with exception of a column of each floor, which is made from brass. During the experiments, the structure is excited with the North-South component of the Mexico City 1985 earthquake, which is fitted in amplitude to be in agreement with the structure and shown in Figure 3. The responses of the shake table and floors are measured through Analog Devices accelerometers, model ADXL203, placed at every floor and at the base. Moreover, the absolute position of each floors is obtained by means of Micro-Epsilon laser sensors, model optoNCDT 1302. These sensors are only used for comparing the displacements and velocities of the floors with their estimates provided by the high gain state observer. Filtering the response of the laser sensors with the filter produces the nominal velocity of the building floors. Filter in (58) has a cut-off frequency of 100 Hz. The parameter identification algorithm and the state observer are implemented in Matlab-Simulink. Data acquisition is carried out through two National Instruments PCI-6221 cards, which communicate with a personal computer by means of the Simulink Desktop Real-Time toolbox. It is worth mentioning that a sampling period of 1 ms is used during the experiments.
The first two natural frequencies of the small scale building are and rad/s, which were obtained from frequency response experiments. It is considered that the first two modes of the structure have a damping factor of 1%, i.e., , based on experimental data and since it is slightly damped. During experiments we have observed that and vary between 0.7% and 1.6%, and this variation depends on the excitation signal. In order to compute the coefficients and of the Rayleigh damping, we decided to fix both damping ratios and to , since this value is between 0.7% and 1.6%. These natural frequencies and damping factors allow obtaining parameters and using (6), which are employed for computing the variables and of parameterization (33).
In order to determine the effectiveness of the cubic spline function in the response reconstruction of the th noninstrumented floor, the following function is computed, which was previously defined in [19]where and are the nominal and estimated absolute acceleration of the th building floor, respectively. Value of function depends on the number of instrumented floors and on their distribution along the building height. Let , , be all possible distributions of instrumented floors of a building with stories. Each distribution produces a set of errors , and the mean value of these errors is denoted as . It is considered that the smaller the value of , the better the performance of the cubic spline functions.
On the other hand, let be the identification error in percentage (%) of the th natural frequency corresponding to estimated building model. Performance index in (60) measures the quality of the identified model by taking into account all errors ; i.e.,The smaller the value of , the better the quality of the model. Since function depends on the integration period of the LIF, it is useful to compute the following terms:
The performance of the high gain state observer is also examined. To this end, function in (62), which also depends on , is computed.The better the reconstruction of the state, the smaller the value of . The following expressions depending on the minimum of function are also computed during the experiments
The goal of the experiments presented in the next subsection is to obtain and compare the value of the performance indexes defined above, when the small scale building is instrumented from one to five floors. A comparison of measured and estimated values of the state will also be included. In all the experiments, the value of the observer gain is fixed to .
7.1. Results with Five Instrumented Floors
This case represents the fully instrumented case, where no cubic splines functions are used, and it is included for comparison purposes. Table 2 shows the values of , , , and for the unique distribution of five instrumented floors, which is named . Moreover, Figures 4(a) and 4(b) show, respectively, the variation of the performance indexes and , where . Note that , indicating that the estimated model with the best quality does not produce the smallest value of . The reason is that term in (43) that depends only on the measurement noise in this case with full instrumentation of the structure affects the performance of state observer, which provokes the fact that the identified model with the best quality does not originate the smallest value . From Figures 4(a) and 4(b), it is possible to see that varies within , whereas function takes values within the interval .
(a) versus
(b) versus
7.2. Results with Four Instrumented Floors
Table 3 presents the parameters , , , , , , and , which are computed using the distributions , . Furthermore, Figures 5(a) and 5(b) depict, respectively, the functions and with respect to . From these figures, it is possible to observe that values of , produced by distributions , , for , are similar. Moreover, functions corresponding to these distributions have a small variation for . On the other hand, the smallest value of is obtained with distribution , but this layout does not produce the smallest values of and . Note that the largest values of and for are computed with distribution , i.e., where the top floor is not instrumented.
(a) versus
(b) versus
7.3. Results with Three Instrumented Floors
Table 4 presents the values of , , , , , , and for distributions , , of three instrumented floors. It is not possible to identify the building model using layout , since the reconstructed absolute acceleration types at noninstrumented floors produce a regressor vector such that covariance matrices in (34) and (35) do not exist. On the other side, Figures 6 and 7 show, respectively, the functions and with respect to . Note that distributions , , which are uniform or approximately uniform along the building height, allow obtaining small values of and , which are close to the ones computed with layouts , , of four instrumented floors. On the other hand, distributions , , that are not uniform over the building height produce the largest values of and ; however, a small value of is computed with .
(a) ,
(b) ,
(a) ,
(b) ,
Figure 8 depicts the absolute acceleration types and and the two noninstrumented floors, which are reconstructed by the cubic shape function in the configuration . It is shown that these responses are close to their nominal value. On the other side, Figure 9 shows the entries , , of the estimated vector provided by the RLS and corresponding to layout . Note that parameters , , converge to a neighborhood around a constant value in approximately 5 s.
(a) Comparison between and
(b) Comparison between and
(a) ,
(b) ,
7.4. Results with Two Instrumented Floors
Values of , , , , , , and for distributions , , are shown in Table 5. Furthermore, Figures 10 and 11 present, respectively, the performance indexes and computed by varying . Layout , with which the building is instrumented at approximately regular intervals, originates the smallest value of and small values of and among the ones shown in Table 5. On the other hand, distributions and that also are approximately uniform along the building height produce large values of but small values of and . It is worth mentioning that configurations , , which are not uniform, generate large values of . Note also that value of , computed with the identified building model corresponding to , is large. Moreover, it is not possible to identify the model of the structure through layouts , .
(a) ,
(b) ,
(a) ,
(b) ,
Figures 12(a) and 12(b) present the displacements estimated by the proposed state observer using layouts and , respectively. Distribution was included for comparison purposes to show a degradation in the quality of the displacement reconstruction when using . In addition, Figure 13 compares the velocities obtained using these same distributions. Note that the quality of the reconstruction of is even worse than that of when using .
(a) State estimate corresponding to the distribution
(b) State estimate corresponding to the distribution
(a) State estimate corresponding to the distribution
(b) State estimate corresponding to the distribution
Note that in the distribution only the acceleration is estimated by means of the cubic spline shape function, whereas in the distribution three acceleration types , , and are reconstructed. Figures 14(a) and 14(b) depict the acceleration obtained with the distributions and , respectively. It is shown that both estimates are close to the nominal acceleration . On the other hand, Figure 15 presents the other two estimated acceleration types and corresponding to the distribution . By comparing Figures 14(b), 15(a), and 15(b), it is concluded that the acceleration that is best reconstructed in the distribution is .
(a) Acceleration corresponding to the distribution
(b) Acceleration corresponding to the distribution
(a) Acceleration obtained with the distribution
(b) Acceleration obtained with the distribution
7.5. Results with One Instrumented Floor
It is not possible to identify the model of the building with all distributions of one instrumented floor, since the reconstructed acceleration responses generate a regressor vector , with which the covariance matrices in (34) and (35) do not exist.
7.6. Parameter Estimates Obtained with Two, Three, Four, and Five Instrumented Floors
Define ) as the th distribution of instrumented floors with its corresponding integration period of the LIF that produces . Table 6 presents the estimated parameters , , by the proposed technique with , , , and . Moreover, this Table shows parameters , , and , computed using those layouts. It is possible to observe that the error in all the estimated natural frequencies and damping factors is less than . Note that for ), ), ), and ), the error of the parameter estimates , , is less than , , , and , respectively. Moreover, for all these distributions, the maximum error obtained by estimating floor mases and stiffness parameters decreases as the number of instrumented floors increases. For example, the floor masses identified with layouts ) and ) have an error less than and , respectively.
7.7. Summary of Experimental Results and Discussion
By comparing the results shown in Sections 7.1–7.6, the following points are concluded:(i)The term tends to decrease as the number of instrumented floors is reduced. Based on this fact and in our experience, the parameter should be selected so that takes a value within the interval , where is the maximum natural frequency of the structure.(ii)Parameter does not depend on the number of instrumented floors. During experiments, term usually takes values close to 20 Hz with either complete or reduced instrumentation.(iii)The quality of the estimated model mainly depends on the sensor location over the structure. When the building is instrumented at regular intervals over its height, then usually the cubic spline shape function yields good results in the reconstruction of the unknown responses and the parameter and state estimators have good performance.(iv)In distributions of sensors that are not uniform along the building height, the cubic spline function produces responses with large errors, and therefore the quality of the estimated model is not guaranteed. In some cases, it is not even possible to identify the model and state of the structure using those responses.(v)As expected, increasing the number of instrumented floors allows decreasing the error of the responses computed by the cubic spline function. This occurs with an increase in the cost of instrumentation. According to our experience, when the ratio is around 0.4–0.6, good results in terms of the performance indexes and and the reproduction of displacements and velocities can be obtained. Values of smaller than 0.4 may yield numerical problems, as shown in the experimental results for the case of one instrumented floor.
Remark 5. The methods described in [1, 3, 6–15] report good results in the parameter estimation of buildings with ratios up to 0.5, 0.33, 0.5, 0.43, 0.5, 0.33, 0.25, 0.5, 0.18, 0.5, 0.5, and 0.5, respectively. Note that the ratio is the most common among them. Moreover, all of these methods operate offline with exception of the techniques presented in [3, 6], and most of them require transforming the building model from continuous one to its discrete time counterpart. Although the ratio , corresponding to the proposed method, is larger than the ratios obtained with the offline techniques described in [9, 10, 12], the proposed technique can be implemented online and can detect parameter changes produced by an anomaly in the structure due to a weak component or failure of elements. Moreover, unlike the methodology in [3], the proposed method does not need measurements of displacements and velocities of the floors, which are more difficult to obtain than floor acceleration measurements.
8. Conclusions
This paper presented a parameter estimation method and a high gain state observer that, respectively, identify the model and state of a shear building using acceleration measurements from only few floor levels and the responses at noninstrumented floors, which are computed through spline cubic shape functions. The building model, obtained by the LS parameter estimation method, is employed by the state observer. The performance of both the observer and the identification method was verified in a five-story experimental structure using one, two, three, four, and five instrumented floors. The experimental results give the following conclusions: it is not possible to estimate the model and state of the structure with only one instrumented floor; in general, the parameter and state estimators produce good results if the layout of recording sensors is uniform or approximately uniform along the height of the building; increasing the number of instrumented floors improves the quality of the estimated model and reduces the state estimation error; if the structure is not instrumented at regular or approximately regular intervals over its height, then a good quality of the identified model is not guaranteed. Even in this case, it may be not possible to identify the model and state of the structure; a reasonable trade-off between estimators performance and the number of instrumented floors occurs when three floors are instrumented. It is worth mentioning that instrumenting two floors also yields good results as long as the distribution is regular.
Appendix
A. Cubic Spline Shape Function Obtained Using the Response at
Let a building with height , stories, be instrumented at its basement and at floors and be divided into subintervals, where each subinterval is delimited by two instrumented floors, as shown in Figure 1. The absolute acceleration at height within the th subinterval , , is estimated through the following cubic shape spline function:For the sake of simplicity the argument of signal and parameters , , , has been omitted in this equation.
Coefficients , , , and are computed by assuming that the cubic polynomials in (A.1) have continuous first and second derivatives with respect to the spatial variable . This assumption allows obtaining the following equations [34]:
First, the unknown parameters of the system of linear equations (A.6) are computed. Then, the constants and are obtained using equations (A.4) and (A.5), respectively. In order to solve the system of equations (A.6), two extra equations are needed, which are deduced from the boundary conditions and . Taking into account these conditions produces
Substituting and , given in Table 1, into (A.7) yields
Substituting into (A.4) and using (A.3) lead to
Expressions (A.6), (A.9), and (A.10) can be rewritten in matrix form as follows:From (A.11), the matrix is computed as follows:where the inverse matrix exists, since is a diagonally dominant matrix. Once matrix is obtained, the parameters and are computed using (A.4) and (A.5).
B. Cubic Spline Shape Function Deduced When the Response at Is Not Available
In this case the building is divided into subintervals, as shown in Figure 1(b), where the subinterval is given byThe absolute acceleration at height within the th subinterval is computed with (A.1), where .
Since the cubic polynomials corresponding to the subintervals and have the same spatial derivative at , the next equality is obtained [34]:
Moreover, the cubic spline function has the following second and third derivatives with respect to at the boundary :
Substituting the boundary conditions , given in Table 1, into (B.3) and (B.4) yieldsMoreover, the boundary condition in Table 1 produces expression (A.10).
Coefficients are computed by solving the set of equations (A.6), (A.10), and (B.5). Subsequently, the parameters and are obtained using (A.4), (B.2) and (A.5), (B.6), respectively.
C. Proof of Proposition 4
Proof. Substituting the gain into term of (47) produceswhere matrices and are defined in (36) and (45), respectively.
Using expression in (46) permits writing the homogeneous part of (C.1) as follows:Define the following equalities:Using these definitions, it is possible to write (C.3) asModal analysis allows expressing (C.5) as the following uncoupled differential equations:where denotes modal coordinate, and parameters , , and are estimates of the modal mass, damping, and stiffness, which are defined aswhere ; variable is the natural mode vector corresponding to the th natural frequency (40) of the estimated building model; i.e., Equation (C.6) is equivalent to the following expression:where and are the modal damping ratios of the identified building model.
Using equalities in (55), it is possible to write (C.9) asThe characteristic equation corresponding to the th differential equation in (C.10) is given byThe roots and , , of (C.11) or equivalently the eigenvalues of in (C.2), are given by expressions (53)–(55). Therefore, point (a) of Proposition 4 has been proved.
On the other hand, substituting equalities in (55) into the characteristic equation (C.11) givesEquation (C.12) can be rewritten asAccording to the root locus technique, the two roots and in (53)–(55) of each characteristic equation in (C.13) traverse the path shown in Figure 16. It is possible to see that root locus has a break-in point where the two roots and return to the real axis. Let be the value of the gain in which the two roots and of the th characteristic equation become real. This gain is computed by considering as zero the imaginary part of and ; i.e.,Thus, increasing from 0 to allows moving the real part of the roots and further and further away from the left side of the real axis. Therefore, if satisfies , then the real part of the roots and , , becomes more and more negative as the gain varies from zero to . Thus, point (b) of Proposition 4 has been proved.
From Figure 16, it is possible to see that roots and of the th characteristic equation converge to the zeros of as , which are given byTherefore, point of Proposition 4 has been proved.
In order to prove point (c), the solution of (48) is used, which is given bywhere is the state transition matrix of system (48).
Since all the eigenvalues and , , of are different, the matrix can be rewritten aswhere columns of matrix are the eigenvectors corresponding to the eigenvalues and , ; moreover, matrix is given by Substituting the equality (C.18) into (C.17) yieldsMatrix (C.20) satisfieswhere is the induced matrix norm, and can be 1, 2, or .
The following inequality follows from (C.21):Applying the matrix norm on both sides of (C.16) and using inequalities (C.21) and (C.23) produceSubstituting the gain (51) into defined in (49) leads towhere and are defined in (57).
Substituting inequality (C.25) into (C.24) and taking the limit of as yield (56). Thus, finally the point (c) of Proposition 4 has been proved.
Nomenclature
| LIF: | Linear Integral Filter |
| LS: | Least Squares |
| RLS: | Recursive Least Squares |
| : | Number of floors of a building |
| : | Mass matrix of the building, kg |
| : | Damping matrix of the building, N·s/m |
| : | Stiffness matrix of the building, N/m |
| : | Displacement vector of the floors relative to the building basement, m |
| : | Velocity vector of the floors relative to the building basement, m/s |
| : | Acceleration vector of the floors relative to the building basement, m/s2 |
| : | Ground acceleration produced by an earthquake, m/s2 |
| : | Vector of ones |
| : | Absolute acceleration vector of the floors, m/s2 |
| : | Rayleigh damping coefficient, 1/s |
| : | Rayleigh damping coefficient, s |
| : | Natural frequency of the th mode, rad/s |
| : | Damping ratio of the th mode, % |
| : | State matrix of a seismically excited building |
| : | Input vector of a seismically excited building |
| : | Output matrix of a seismically excited building |
| : | State of a seismically excited building |
| : | Number of instrumented floors |
| : | Localization matrix of accelerometers |
| : | Matrix composed of acceleration of floors relative to the building basement, m/s2 |
| : | Stiffness/mass vector to be identified, N/(kg·m) |
| : | Superscript used in estimated parameters or signals |
| : | LIF representing integrals over finite time intervals of |
| : | Time integration window length of the LIF, s |
| : | Sampling period, s |
| : | Laplace operator |
| : | Regressor vector |
| : | Output of the parameterization model |
| : | th-order LIF |
| : | Cut-off frequency of the filter , rad/s |
| : | Cut-off frequency of the filter , Hz |
| , : | Band pass filters |
| : | Building height |
| : | Height of the th instrumented floor |
| : | Absolute acceleration of the th instrumented floor, m/s2 |
| : | Absolute acceleration at height of the building, m/s2 |
| : | Absolute acceleration at the th noninstrumented floor, m/s2 |
| : | Height at the th noninstrumented floor |
| : | Subinterval delimited by two instrumented floors |
| : | Coefficients of the th cubic polynomial |
| : | Error between estimated and nominal relative acceleration, m/s2 |
| : | Error between estimated and nominal absolute acceleration, m/s2 |
| : | Matrix composed of errors between estimated and nominal relative acceleration, m/s2 |
| : | Signal depending on the acceleration estimation errors |
| : | Number of samples used by the LS |
| : | Covariance matrix of the LS or RLS |
| : | State observer gain matrix |
| : | Multiplying factor of the state observer gain matrix |
| : | th distribution of instrumented floors |
| : | Metric to determine the effectiveness in the reconstruction of the th noninstrumented floor |
| : | Mean value of a set of errors of a distribution of instrumented floors |
| : | Identification error in percentage of the th estimated natural frequency, % |
| : | Performance index that measures the quality of the identified model |
| : | Minimum value of |
| : | Time integration window length that produces the minimum value of , s |
| : | The reciprocal of , Hz |
| : | Performance of the high gain state observer |
| : | Minimum value of function |
| : | Time integration window length that produces the minimum value of , s |
| : | The reciprocal of , Hz. |
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
Antonio Concha acknowledges Programa para el Desarrollo Profesional Docente (PRODEP-SEP) for supporting this research. Luis Alvarez-Icaza acknowledges support from UNAM-PAPIIT Grant IN104218.