Shock and Vibration

Volume 2018, Article ID 5206968, 21 pages

https://doi.org/10.1155/2018/5206968

## Parameter and State Estimation of Shear Buildings Using Spline Interpolation and Linear Integral Filters

^{1}Facultad de Ingeniería Mecánica y Eléctrica, Universidad de Colima, 28400 Coquimatlán, COL, Mexico^{2}Instituto de Ingeniería, Universidad Nacional Autónoma de México, 04510 Coyoacán, CDMX, Mexico

Correspondence should be addressed to Luis Alvarez-Icaza; xm.manu.negnii.samup@ravla

Received 25 February 2018; Accepted 18 April 2018; Published 19 July 2018

Academic Editor: Daniele Baraldi

Copyright © 2018 Antonio Concha and Luis Alvarez-Icaza. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 .