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

Authors: Antonio Concha; Luis Alvarez-Icaza
Journal: Shock and Vibration
 (2018)

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## 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.


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## Body

## 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 P of instrumented floors should satisfy P≥n and P≥(1+4n-3)/2 if the floor masses at sensor locations are known and if the total mass of the structure is known, respectively, where n 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(1) an identification technique to identify the parameters of the model of a seismically excited shear building using a limited number of responses and (2) 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: (1) assuming that building has Rayleigh damping; (2) containing a vector whose entries are the stiffness/mass ratios of the building, which are estimated through a Least Squares algorithm; and (3) 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. Sections2 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 withn stories subjected to earthquake or base motion is described through the following mathematical model [27]:(1)Mx¨t+lx¨gt+Cx˙t+Kxt=0(2)xt=x1t,…,xntT,x˙t=x˙1t,…,x˙ntTx¨t=x¨1t,…,x¨ntT,l=1,1,…,1T(3)x¨at=x¨t+lx¨gt=x¨a1t,…,x¨antT,where variables xi(t), x˙i(t), and x¨i(t) are, respectively, the displacement, velocity, and acceleration of the ith floor, which are measured with respect to the basement. Signal x¨ai(t) represents the absolute acceleration at the ith floor, and x¨g(t) is the ground acceleration produced by earthquake. Moreover, M and K are the mass and stiffness matrices, respectively, which are defined as (4)M=m10⋯00m2⋯0⋮⋮⋱⋮00⋯mn,K=k1+k2-k2⋯0-k2k2+k3⋯0⋮⋮⋱⋮00⋯kn The entry ki(i=1,2,…,n) is the column lateral stiffness between the ith and (i-1)th floors.The building has Rayleigh damping that is represented through the following matrixC:(5)C=a0M+a1KParameters a0 and a1 are constant, and they are computed as [27](6)121ωiωi1ωjωja0a1=ζiζj,where ζv and ωv, v=i,j, are the damping ratio and natural frequency of the vth structural mode, respectively.Remark 1.
Computing the constantsa0 and a1 in (6) requires knowledge of the parameters ωi, ωj, ζi, and ζj that corresponds to the ith and jth structural modes. Natural frequencies ωi and ωj 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 ζi and ζj 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 Table11.2.1 from [27], which depend on the type and condition of the structure.The state space model corresponding to the differential equation (1) is given by(7)η˙t=Aηt+Bx¨gt(8)A=On×nIn×n-M-1K-M-1C,B=On×1-l(9)ηt=xTt,x˙TtT=x1t,x2t,…,xnt,x˙1t,x˙2t,…,x˙ntT,where Ip×r and Op×r are the identity and zero matrices with size p×r, respectively.The output equation of the state space model depends on the absolute acceleration of the instrumented floors. LetP be the number of instrumented floors, and let i=1,2,…,P be the ith instrumented floor numbered from bottom to top of the building. Then, the output equation is given by(10)yt=Γx¨at=Γx¨t+lx¨gt=ΓDηt,D=-M-1K,-M-1C,where Γ∈RPxn is the localization matrix of the accelerometers, whose entries are defined as(11)Γi,j=1ifthejthflooristheithinstrumentedstory0otherwise


## 3. Model Parameterization

Assume first that all the stories are instrumented; it means thatΓ=Inxn; then substituting (5) into (1) yields(12)x¨t+lx¨gt=-M-1a0M+a1Kx˙t-M-1KxtEquation (12) can be rewritten as(13)x¨at+a0x˙t=-a1M-1Kx˙t-M-1KxtDefine the following equalities:(14)M-1Kx˙t=u˙tθ,M-1Kxt=utθ,where u˙(t) is the time derivative of u(t) and(15)θ=k1m1k2m1k2m2⋯knmnT=θ1θ2θ3⋯θ2n-1Tut=-x1tx1t-x2t0⋯000x2t-x1t⋯0⋮⋮⋮⋮⋮000⋯xnt-xn-1tUsing equalities in (14) leads to the following expression: (16)x¨at+a0x˙t=a1u˙t+utθIn 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 obtaining(17)I5x¨a3t+a0x¨2t=I5a1u¨2t+u¨1tθThe superindex (i) denotes the ith time derivative of the corresponding signal; and I5{·} is a Linear Integral Filter (LIF). The general definition a LIF is given by(18)Iαρt=1δα-1∫t-δt∫τ1-δτ1⋯∫τα-1-δτα-1ρταdτα⋯dτ1,where α is the number of integrations, and δ is the time integration window length defined as (19)δ=ħTs, where ħ>0, and Ts is the sampling period of ρ.The Laplace transform ofIα{ρ(t)} in (18) is given by [17](20)LIαρt=δGαsLρt(21)Gαs=1-e-δsδsα,Gαjω=sin⁡πω/ωcπω/ωcα,ωc=2πδ,fc=1δTerms ωc and fc determine the bandwidth of the αth-order low-pass filter Gα(s) in rad/s and Hz, respectively.Equation (17) is equivalent to the following parameterization, which will be employed for parameter identification purposes:(22)χt=ϕtθ(23)χt=1δ3I2∑j=033j-1jx¨at-jδ+a0δ2I3∑j=022j-1jx¨t-jδ(24)ϕt=a1δ2I3∑j=022j-1ju¨t-jδ+1δI4∑j=011j-1ju¨t-jδIt is important to mention that variablesχ(t) and ϕ(t) can be written in the Laplace domain as(25)Lχt=H3sLx¨at+a0H2sLx¨tLϕt=a1H2sLu¨t+H1sLu¨t,where(26)H1s=δsG5s,H2s=δs2G5s,H3s=δs3G5sRemark 2.
The filtersHi(s), i=1,2,3, 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 signalsx¨a1(t),…,x¨an(t), which appear in variables x¨(t), x¨a(t), and u¨(t) of parameterization (22), are available. In order to implement x¨(t), x¨a(t) and u¨(t), 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 heightH and n stories that is instrumented at its basement and at P floors, as shown in Figure 1. Moreover, let h0=0 and r0(t)=x¨g(t) be the height and acceleration response at the basement, respectively. Similarly, terms hp and rp(t), p=1,2,…,P, are the height and absolute acceleration at the pth instrumented floor, respectively, where rp(t)∈x¨a(t).Figure 1
Building with heightH instrumented in P floor levels.
(a)
Building instrumented at the top floor (b)
Building noninstrumented at the top floorLetx¨a(h,t) be the absolute acceleration of the building at height h. Using (3) yields x¨a(0,t)=x¨g(t) and x¨a(H,t)=x¨na(t). In addition, let h∗ be the height of the wth noninstrumented floor that is located within the subinterval Δp=hp+1-hp delimited by two instrumented floors with heights hp and hp+1, where hp+1>hp. The response at this noninstrumented floor is given by x¨a(h∗,t), or, equivalently, x¨a(h∗,t)=x¨aw(t) according to definition (3). An estimate x¨^a(h∗,t)=x¨^aw(t) of x¨a(h∗,t)=x¨wa(t) is computed through the following cubic spline shape function:(27)x¨^awt=x¨^ah∗,t=apt+bpth∗-hp+dph∗-hp2+epth∗-hp3,h∗∈hp,hp+1,where ap(t), bp(t), dp(t), and ep(t) are the coefficients of the pth cubic polynomial, which are computed at every sampling instant from continuity of the spline function, from responses rp(t), p=0,1,2,…,P, 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 h. Appendices A and B present the cubic spline function obtained if the acceleration response at the top floor is available or not, respectively.Table 1
Boundary conditions for a shear building.
Boundary Absolute accelerationx¨a(t)=x¨(t)+x¨g(t) h=0 x¨a(0,t)=x¨g(t) x ¨ a ′ ( 0 , t ) = 0 h=H x ¨ a ′ ′ ( H , t ) = 0 x ¨ a ′ ′ ′ ( H , t ) = 0Once absolute accelerationx¨^aw(t) of the wth noninstrumented floor has been obtained, it is possible to estimate its relative acceleration x¨^w(t) through the expression x¨^w(t)=x¨^aw(t)-x¨g(t). Then, the unavailable responses x¨w(t) and x¨aw(t) in variables x¨(t), x¨a(t) and u¨(t) are replaced by their estimates x¨^w(t) and x¨^aw(t), respectively. From now on, variables constructed with recorded and reconstructed responses are denoted as x¨^a(t), x¨^(t), and u¨^(t), which are given by(28)x¨^t=x¨t+εtx¨^at=x¨at+εatu¨^t=u¨t+εut,where terms ε(t), εa(t), and εu(t) depend on the error of the reconstructed responses and on the noise of the recorded responses.


## 5. Parameter Estimation of the Building

Substituting signalsx¨^(t), x¨^a(t), and u¨^(t) given in (28) into (22)–(24) yields(29)χ^t=ϕ^tθ+I5λt,where variables χ^(t) and ϕ^(t) have the same structure as χ(t) and ϕ(t), respectively. However, χ^(t) and ϕ^(t) contain the terms x¨^a(t), x¨^(t), and u¨^(t) instead of x¨a(t), x¨(t), and u¨(t). On the other hand, vector λ(t) depends on ε, εa, and εu. The Laplace transform of λ(t) is given by(30)Λs=Lλt=s3Lεa+s2a0Lε-a1Lεuθ-sLεuθEmploying (20) and (30) allows obtaining the next Laplace transform of I5[λ(t)](31)LI5λt=H3sLεa+H2sa0Lε-a1Lεuθ-H1sLεuθ,where Hi(s), i=1,2,3, were previously defined in (26).Expression (29) can be rewritten as(32)χ^kTs=ϕ^kTsθ+I5λkTs,where t=kTs, k=0,1,2,…, are the sampling instants of signals χ^(t) and ϕ^(t). Omitting Ts in (32) leads to(33)χ^k=ϕ^kθ+I5λkIn order to estimate the parameter vectorθ in (33), the Least Squares (LS) algorithm is employed, which is defined as [31](34)θ^=PN∑k=0Nϕ^Tkχ^k,PN=∑k=0Nϕ^Tkϕ^k-1,where P is the covariance matrix; N is the number of samples of χ^ and ϕ^. Note that vector θ^ can be computed only if matrix P exists.On the other hand,θ can also be recursively identified since the responses at noninstrumented floor levels can be computed at every sampling period Ts. The recursive version of the LS, denoted as RLS, is given by(35)θ^k=θ^k-1+LkϵkLk=Pk-1ϕ^TkβIn×n+ϕ^kPk-1ϕ^kT-1Pk=Pk-1-Lkϕ^kPk-1βϵk=χ^k-ϕ^kθ^k,where β is the forgetting factor such that 0<β≤1; moreover, ϵ(k) is the output estimation error.Proposition 3.
Suppose that input signalx¨g(t) is persistently exciting at least of order b=2n-1, where b is the number of parameters of θ to be estimated, then vector norm θ~ of the parameter estimation error θ~, defined as θ~=θ-θ^, is bounded. Moreover, the smaller the perturbation term I5[λ(kTs)] in (33), the smaller the value of θ~.Proof.
It is given in books [31, 32].


### 5.1. Estimation of Modal Parameters

Once that vectorθ^ 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 A^, which is an estimate of matrix A given in (8), is constructed:(36)A^=On×nIn×n-M-1K^-M-1C^(37)M-1K^=θ^1+θ^2-θ^2⋯0-θ^3θ^3+θ^4⋯0⋮⋮⋱⋮00⋯θ^2n-1(38)M-1C^=a0In×n+a1M-1K^,where parameters θ^i, i=1,2,…,2n-1, are the entries of vector θ^. Note that (38) is deduced from (5).The eigenvalues of the matrixA^ in (36) are given by(39)λi,1=-σi+jϖi=-ζ^iω^i+jω^i1-ζ^2,λi,2=-σi-jϖi=-ζ^iω^i-jω^i1-ζ^2,where σi=ζ^iω^i and ϖi=ω^i1-ζ^2. Moreover, ω^i and ζ^i, i=1,2,…,n, are, respectively, the estimates of the natural frequency ωi and damping factor ζi corresponding to the ith mode of building model (1).From (39), the following equations for computing the parameters ω^i and ζ^i are obtained:(40)ω^i=σi2+ϖi2,ζ^i=σiω^i,i=1,2,…,n


### 5.2. Estimation of MatricesM, K, and C

Assume that massm1 of the building is known; then matrix C^ and the entries of matrices M^ and K^ are given by(41)k^1=m^1θ^1,k^i=m^i-1θ^2i-2,m^i=k^iθ^2i-1,i=2,3,…,nC^=a0M^+a1K^A similar procedure can be carried out if another floor mass is known instead of m1.


## 5.1. Estimation of Modal Parameters

Once that vectorθ^ 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 A^, which is an estimate of matrix A given in (8), is constructed:(36)A^=On×nIn×n-M-1K^-M-1C^(37)M-1K^=θ^1+θ^2-θ^2⋯0-θ^3θ^3+θ^4⋯0⋮⋮⋱⋮00⋯θ^2n-1(38)M-1C^=a0In×n+a1M-1K^,where parameters θ^i, i=1,2,…,2n-1, are the entries of vector θ^. Note that (38) is deduced from (5).The eigenvalues of the matrixA^ in (36) are given by(39)λi,1=-σi+jϖi=-ζ^iω^i+jω^i1-ζ^2,λi,2=-σi-jϖi=-ζ^iω^i-jω^i1-ζ^2,where σi=ζ^iω^i and ϖi=ω^i1-ζ^2. Moreover, ω^i and ζ^i, i=1,2,…,n, are, respectively, the estimates of the natural frequency ωi and damping factor ζi corresponding to the ith mode of building model (1).From (39), the following equations for computing the parameters ω^i and ζ^i are obtained:(40)ω^i=σi2+ϖi2,ζ^i=σiω^i,i=1,2,…,n


## 5.2. Estimation of MatricesM, K, and C

Assume that massm1 of the building is known; then matrix C^ and the entries of matrices M^ and K^ are given by(41)k^1=m^1θ^1,k^i=m^i-1θ^2i-2,m^i=k^iθ^2i-1,i=2,3,…,nC^=a0M^+a1K^A similar procedure can be carried out if another floor mass is known instead of m1.


## 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 by(42)η^˙t=A^η^t+Bx¨gt+Lψ~t,ψ~t=x¨^at-y^ot(43)x¨^at=x¨at+εat=Dηt+εat,y^ot=D^η^t(44)η^=x^T,x˙^TT=x^1,x^2,…,x^n,x˙^1,x˙^2,…,x˙^nT,where η^ is an estimate of η, L is the observer gain matrix, variable x¨^a is established in (28), term y^o is the absolute acceleration estimated by the observer, matrix A^ is presented in (36), and matrix D^, which is an estimate of D in (10), is defined as(45)D^=M-1K^,M-1C^with M-1K^ and M-1C^ shown in (37) and (38), respectively.Define the state estimation errorη~(t) as follows:(46)η~t=ηt-η^t=xtx˙t-x^tx˙^t=x~tx˙~t=x~1,x~2,…,x~n,x˙~1,x˙~2,…,x˙~nTThen, the state estimation error dynamics η~˙(t) is given by(47)η~˙t=η˙t-η^˙t=A-LDηt+A^-LD^η~t-Lεat=A-LDηt+A^-LD^ηt-η^t-Lεat=A^-LD^η~t+Ξt(48)=A∗η~t+Ξtwith(49)A∗=A^-LD^,Ξt=A~-LD~ηt-Lεat(50)A~=A-A^,D~=D-D^Proposition 4.
LetL be the gain of the state observer defined as(51)L=On×n-γIn×nT(1)
Ifγ satisfies(52)0≤γ≤κ=min⁡ϱ1,ϱ2,…,ϱn,ϱi=1ζ^i2-1,i=1,2,…,n,whereζ^i is presented in (40), and min⁡· denotes the smallest value of the set ·, then, one has the following:
(a)
The eigenvalues of matrixA∗ in (48) are given by(53)λi,1∗=-σi∗+jϖi∗,λi,2∗=-σi∗-jϖi∗(54)σi∗=ζi∗ωi∗ϖi∗=ωi∗1-ζ∗2(55)ωi∗=ω^i1+γ,ζi∗=ζ^i1+γ,wherei=1,2,…,n and ω^i and ζ^i are the natural frequency and damping factor of the ith mode of the estimated building model, which are computed using (40).(b)
Increasing the value of gainγ in (51)-(52) allows the real part of the eigenvalues of matrix A∗ to be more negative.(c)
Norm of the estimation errorη~ is bounded, and when t→∞ it satisfies(56)η^t≤μζ^•ω^•Eηt+μγεat1+γζ^•ω^•(57)E=On×n11+γIn×n-M-1K~-M-1C~,M-1K~=M-1K-M-1K^,M-1C~=M-1C-M-1C^,μ=MM-1,ζ^•ω^•=min1≤i≤nζ^iω^i,whereM is a matrix, whose columns are the eigenvectors of matrix A∗.(2)
Ifγ→∞, then the eigenvalues of A∗ approach λi,1∗=-ω^i/(2ζ^i) and λi,2∗=-∞, i=1,2,…,n.Proof.
See AppendixC.


### 6.1. Attenuation of the State Estimation Error

The normη^(t) of the state estimation error can be reduced as follows:(i)
Increaseγ in order to decrease E and to obtain a fast response for the observer. Based on our experience, good results in the state estimation are obtained using a gain γ between κ/10 and κ, where κ was defined in (52).(ii)
ReduceM-1K~ and M-1C~, which allows decreasing E. According to Proposition 3, the smaller the term I5[λ] in (33) is, the smaller the parametric error norms M-1K~ and M-1C~ are. Signal I5[λ] is filtered through the filters H1(s), H2(s), and H3(s) 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 I5[λ] and norms M-1K~ and M-1C~.(iii)
Instrument the building at regular intervals over its height, which permits reducingεa [19, 33]. These references also show that increasing the total number of instrumented floors decreases the value of εa.(iv)
Attenuate the measurement noise corrupting the state observer. To achieve this, vectorx¨^a(t) in (28) is replaced by the following filtered vector x¨^af(t) in variable ψ~ of the state observer (42):(58)x¨^aft=L-1F1s∗x¨^at,where∗ is the convolution operator, and F1(s) is a fourth-order low-pass Butterworth filter, whose cut-off frequency is appropriately chosen.


## 6.1. Attenuation of the State Estimation Error

The normη^(t) of the state estimation error can be reduced as follows:(i)
Increaseγ in order to decrease E and to obtain a fast response for the observer. Based on our experience, good results in the state estimation are obtained using a gain γ between κ/10 and κ, where κ was defined in (52).(ii)
ReduceM-1K~ and M-1C~, which allows decreasing E. According to Proposition 3, the smaller the term I5[λ] in (33) is, the smaller the parametric error norms M-1K~ and M-1C~ are. Signal I5[λ] is filtered through the filters H1(s), H2(s), and H3(s) 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 I5[λ] and norms M-1K~ and M-1C~.(iii)
Instrument the building at regular intervals over its height, which permits reducingεa [19, 33]. These references also show that increasing the total number of instrumented floors decreases the value of εa.(iv)
Attenuate the measurement noise corrupting the state observer. To achieve this, vectorx¨^a(t) in (28) is replaced by the following filtered vector x¨^af(t) in variable ψ~ of the state observer (42):(58)x¨^aft=L-1F1s∗x¨^at,where∗ is the convolution operator, and F1(s) is a fourth-order low-pass Butterworth filter, whose cut-off frequency is appropriately chosen.


## 7. Experimental Results

Figure2 depicts a five-story small scale building with dimensions 65×55×175.5 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 (75π)2s/(s+75π)2 produces the nominal velocity of the building floors. Filter F1(s) 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.Figure 2
Experimental structure.Figure 3
Excitation of the experimental structure.The first two natural frequencies of the small scale building areω1=11.07 and ω2=31.8rad/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., ζ1=ζ2=0.01, based on experimental data and since it is slightly damped. During experiments we have observed that ζ1 and ζ2 vary between 0.7% and 1.6%, and this variation depends on the excitation signal. In order to compute the coefficients a0 and a1 of the Rayleigh damping, we decided to fix both damping ratios ζ1 and ζ2 to 1%, since this value is between 0.7% and 1.6%. These natural frequencies and damping factors allow obtaining parameters a1=0.1642 and a2=0.0005 using (6), which are employed for computing the variables χ^ and ϕ^ of parameterization (33).In order to determine the effectiveness of the cubic spline function in the response reconstruction of thewth noninstrumented floor, the following function Ew is computed, which was previously defined in [19](59)Ew=∑k=0Px¨awk-x¨^awk2∑k=0Nx¨aw2k,where x¨wa and x¨^wa are the nominal and estimated absolute acceleration of the wth building floor, respectively. Value of function Ew depends on the number of instrumented floors and on their distribution along the building height. Let ΠjP, j=1,2,…,ι;ι=(n!/[P!(n-P)!]), be all possible distributions of P instrumented floors of a building with n stories. Each distribution produces a set of errors Ew, and the mean value of these errors is denoted as E¯. It is considered that the smaller the value of E¯, the better the performance of the cubic spline functions.On the other hand, letω~i be the identification error in percentage (%) of the ith natural frequency ω^i corresponding to estimated building model. Performance index Eω~ in (60) measures the quality of the identified model by taking into account all errors ω~i; i.e.,(60)Eω~δ=∑i=15ω~i2,ω~i%=ω^i-ωiωi×100The smaller the value of Eω~, the better the quality of the model. Since function Eω~ depends on the integration period δ of the LIF, it is useful to compute the following terms:(61)Eω~∗=minδ∈δmin,δmaxEω~δ,δ∗=arg⁡minδ∈δmin,δmax⁡Eω~δ,fc∗=1δ∗The performance of the high gain state observer is also examined. To this end, functionEx~(δ) in (62), which also depends on δ, is computed.(62)Ex~δ=∑i=15x~i+x˙~i,x~i=∑j=1Nx~i2,x˙~i=∑j=1Nx˙~i2The better the reconstruction of the state, the smaller the value of Ex~(δ). The following expressions depending on the minimum of function Ex~(δ) are also computed during the experiments(63)Ex~∘=minδ∈δmin,δmaxEx~δ,δ∘=arg⁡minδ∈δmin,δmax⁡Ex~δ,fc∘=1δ∘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 gainγ is fixed to γ=1000.


### 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. Table2 shows the values of Eω~∗, δ∗, Ex~∘, and δ∘ for the unique distribution of five instrumented floors, which is named Π15. Moreover, Figures 4(a) and 4(b) show, respectively, the variation of the performance indexes Eω~(δ) and Ex~(δ), where δ∈[0.01,0.12]. Note that δ∗≠δ∘, indicating that the estimated model with the best quality does not produce the smallest value of Ex~(δ). The reason is that term εa 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 Ex~∘. From Figures 4(a) and 4(b), it is possible to see that Eω~(δ) varies within [1.96,29.47], whereas function Ex~(δ) takes values within the interval [6.37,14.84].Table 2
Results obtained when all floors are instrumented.
Πj5 Instrumented floors Non-instrumented floors E- Eω~∗ δ ∗ [s] f c ∗ [Hz] Ex~∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 2 3 4 5 – – 1.96 0.035 28.57 6.37 0.055 18.18Figure 4
FunctionsEω~(δ) and Ex~(δ) computed when all the floors are instrumented.
(a)
E ω ~ δ versus  δ (b)
E x ~ ( δ ) versus δ


### 7.2. Results with Four Instrumented Floors

Table3 presents the parameters E¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘, which are computed using the distributions Πj4, j=1,2,…,5. Furthermore, Figures 5(a) and 5(b) depict, respectively, the functions Eω~ and Ex~ with respect to δ. From these figures, it is possible to observe that values of Eω~(δ), produced by distributions Πk4, j=1,2,…,4, for δ∈[0.02,0.05], are similar. Moreover, functions Ex~(δ) corresponding to these distributions have a small variation for δ∈[0.05,0.1]. On the other hand, the smallest value of E¯ is obtained with distribution Π34, but this layout does not produce the smallest values of Eω~∗ and Ex~∘. Note that the largest values of Eω~(δ) and Ex~(δ) for δ∈[0.02,0.06] are computed with distribution Π54, i.e., where the top floor is not instrumented.Table 3
Results with four instrumented floors.
Π j 4, j Instrumented floors Non-instrumented floors E- Eω~∗ δ ∗ [s] f c ∗ [Hz] Ex~∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 2 4 5 3 0.18 3.50 0.04 25 11.64 0.048 20.83 2 0 1 3 4 5 2 0.27 3.59 0.038 26.32 9.45 0.058 17.24 3 0 1 2 3 5 4 0.16 4.18 0.04 25 13.38 0.048 20.83 4 0 2 3 4 5 1 0.45 5.79 0.044 22.73 7.50 0.062 16.13 5 0 1 2 3 4 5 0.32 12.32 0.068 14.71 26.07 0.048 20.83Figure 5
FunctionsEω~(δ) and Ex~(δ) computed with four instrumented floors.
(a)
E ω ~ δ versus  δ (b)
E x ~ ( δ ) versus δ


### 7.3. Results with Three Instrumented Floors

Table4 presents the values of E¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘ for distributions Πj3, j=1,2,…,9, of three instrumented floors. It is not possible to identify the building model using layout Π103, since the reconstructed absolute acceleration types at noninstrumented floors produce a regressor vector ϕ^ such that covariance matrices P in (34) and (35) do not exist. On the other side, Figures 6 and 7 show, respectively, the functions Eω~ and Ex~ with respect to δ. Note that distributions Πj3, j=1,2,3, which are uniform or approximately uniform along the building height, allow obtaining small values of Eω~∗ and Ex~∘, which are close to the ones computed with layouts Πj4, j=1,2,3, of four instrumented floors. On the other hand, distributions Πk3, k=7,8,9, that are not uniform over the building height produce the largest values of E¯ and Eω~∗; however, a small value of Ex~∘ is computed with Π93.Table 4
Results with three instrumented floors.
Πj3,j Instrumented floors Non-instrumented floors E - E ω ~ ∗ δ ∗ [s] f c ∗ [Hz] E x ~ ∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 3 5 2 4 0.48 5.17 0.046 21.74 13.88 0.054 18.52 2 0 2 4 5 1 3 0.68 5.21 0.048 20.83 11.98 0.05 20 3 0 2 3 5 1 4 0.63 6.08 0.046 21.74 14.63 0.056 17.86 4 0 1 2 5 3 4 0.53 8.55 0.054 18.52 25.4 0.048 20.83 5 0 1 3 4 2 5 0.61 9.03 0.05 20 26.27 0.046 21.74 6 0 1 2 4 3 5 0.55 9.98 0.05 20 21.83 0.044 22.73 7 0 1 4 5 2 3 0.74 10.98 0.052 19.23 18.71 0.038 26.32 8 0 2 3 4 1 5 0.78 13.15 0.07 14.29 25.88 0.05 20 9 0 3 4 5 1 2 0.98 14.57 0.06 16.67 12.71 0.064 15.63 10 0 1 2 3 4 5 0.96 – – – – – –Figure 6
FunctionEω~(δ) calculated with three instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figure 7
FunctionEx~(δ) computed with three instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figure8 depicts the absolute acceleration types x¨^a2 and x¨^a4 and the two noninstrumented floors, which are reconstructed by the cubic shape function in the configuration Π13. It is shown that these responses are close to their nominal value. On the other side, Figure 9 shows the entries θ^i, i=1,2,…,9, of the estimated vector θ^ provided by the RLS and corresponding to layout Π13. Note that parameters θ^i, i=1,2,…,9, converge to a neighborhood around a constant value in approximately 5 s.Figure 8
Reconstructed signalsx¨^2a and x¨^4a corresponding to the distribution Π13.
(a)
Comparison betweenx¨2a and x¨^2a (b)
Comparison betweenx¨4a and x¨^4aFigure 9
Estimated vectorθ^ produced by the RLS and corresponding to the configuration Π13.
(a)
θ ^ i, i=1,2,3,4,5 (b)
θ ^ i, i=6,7,8,9


### 7.4. Results with Two Instrumented Floors

Values ofE¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘ for distributions Πj2, j=1,2,…,7, are shown in Table 5. Furthermore, Figures 10 and 11 present, respectively, the performance indexes Eω~ and Ex~ computed by varying δ. Layout Π12, with which the building is instrumented at approximately regular intervals, originates the smallest value of Eω~∗ and small values of E¯ and Ex~∘ among the ones shown in Table 5. On the other hand, distributions Π52 and Π62 that also are approximately uniform along the building height produce large values of Eω~∗ but small values of E¯ and Ex~∘. It is worth mentioning that configurations Πj2, j=7,8,9,10, which are not uniform, generate large values of E¯. Note also that value of Eω~∗, computed with the identified building model corresponding to Π72, is large. Moreover, it is not possible to identify the model of the structure through layouts Πj2, j=8,9,10.Table 5
Results with two instrumented floors.
Π j 2 j Instrumented floors Non-instrumented floors E - E ω ~ ∗ δ ∗ [s] f c ∗ [Hz] E x ~ ∘ δ ∘ [s] f c ∘ [Hz] 1 0 2 4 1 3 5 1.01 5.40 0.054 18.52 20.7 0.042 23.81 2 0 1 4 2 3 5 1.14 12.46 0.074 13.52 28.42 0.048 20.83 3 0 1 5 2 3 4 1.23 14.68 0.084 11.90 29.29 0.04 25 4 0 3 4 1 2 5 1.28 18.35 0.06 16.67 25.37 0.06 16.67 5 0 2 5 1 3 4 0.99 19.05 0.058 17.24 23.83 0.046 21.74 6 0 3 5 1 2 4 1.14 33.9 0.064 15.63 16.12 0.048 20.83 7 0 4 5 1 2 3 1.47 38.54 0.06 16.67 22.71 0.05 20 8 0 2 3 1 4 5 1.41 – – - – – – 9 0 1 3 2 4 5 1.39 – – - – – – 10 0 1 2 3 4 5 2.24 – – – – – –Figure 10
FunctionEω~(δ) computed with two instrumented floors.
(a)
Π k 2, k=1,2,3,4,5 (b)
Π k 2, k=6,7Figure 11
FunctionEx~(δ) computed with two instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figures12(a) and 12(b) present the displacements x^5 estimated by the proposed state observer using layouts Π14 and Π12, respectively. Distribution Π14 was included for comparison purposes to show a degradation in the quality of the displacement reconstruction when using Π12. In addition, Figure 13 compares the velocities x˙^5 obtained using these same distributions. Note that the quality of the reconstruction of x˙^5 is even worse than that of x^5 when using Π12.Figure 12
Estimates produced by the state observer in the distributionsΠ14 and Π12.
(a)
State estimatex^5 corresponding to the distribution Π14 (b)
State estimatex^5 corresponding to the distribution Π12Figure 13
Estimates produced by the state observer in the distributionsΠ14 and Π12.
(a)
State estimatex˙^5 corresponding to the distribution Π14 (b)
State estimatex˙^5 corresponding to the distribution Π12Note that in the distributionΠ14 only the acceleration x¨3a is estimated by means of the cubic spline shape function, whereas in the distribution Π12 three acceleration types x¨1a, x¨3a, and x¨5a are reconstructed. Figures 14(a) and 14(b) depict the acceleration x¨^3a obtained with the distributions Π14 and Π12, respectively. It is shown that both estimates are close to the nominal acceleration x¨3a. On the other hand, Figure 15 presents the other two estimated acceleration types x¨^1a and x¨^5a corresponding to the distribution Π12. By comparing Figures 14(b), 15(a), and 15(b), it is concluded that the acceleration that is best reconstructed in the distribution Π12 is x¨3a.Figure 14
Accelerationx¨^3a obtained with the distributions Π14 and Π12.
(a)
Accelerationx¨^3a corresponding to the distribution Π14 (b)
Accelerationx¨^3a corresponding to the distribution Π12Figure 15
Accelerationx¨^1a and acceleration x¨^5a corresponding to the distributions Π14 and Π12.
(a)
Accelerationx¨^1a obtained with the distribution Π12 (b)
Accelerationx¨^5a obtained with the distribution Π12


### 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 vectorϕ^, with which the covariance matrices P in (34) and (35) do not exist.


### 7.6. Parameter Estimates Obtained with Two, Three, Four, and Five Instrumented Floors

Define(ΠjP,δ∗) as the jth distribution of P instrumented floors with its corresponding integration period δ∗ of the LIF that produces Eω~∗. Table 6 presents the estimated parameters θ^i, i=1,2,…,9, by the proposed technique with (Π12,δ∗), (Π13,δ∗), (Π14,δ∗), and (Π15,δ∗). Moreover, this Table shows parameters ω^r, ζ^r, m^r and k^r, r=1,2,…,5 computed using those layouts. It is possible to observe that the error in all the estimated natural frequencies ω^r and damping factors ζ^r is less than 4%. Note that for (Π15,δ∗), (Π14,δ∗), (Π13,δ∗), and (Π12,δ∗), the error of the parameter estimates θ^i, i=1,2,…,9, is less than 10%, 20%, 25%, and 73%, 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 (Π13,δ∗) and (Π14,δ∗) have an error less than 18% and 8%, respectively.Table 6
Parameter estimates corresponding to the distributionsΠ12, Π13, Π14, and Π15 of two, three, four, and five floors.
Nominal value Unit Estimate(Π15,δ∗) Error% Estimates(Π14,δ∗) Error% Estimates(Π13,δ∗) Error% Estimates(Π12,δ∗) Error% θ 1 = 1708.3 1/s2 1648.0 3.53 1629.0 4.64 1644.8 3.72 2940.5 72.13 θ2=1144.3 1/s2 1134.4 0.86 1105.5 3.39 1121.8 1.96 1608.1 40.53 θ3=1340.9 1/s2 1319.1 1.63 1262.9 5.82 1462.1 9.04 1104.4 17.64 θ4=1340.9 1/s2 1337.7 0.24 1343.2 0.17 1504.3 12.19 1230.3 8.25 θ5=1340.9 1/s2 1257.5 6.22 1526.8 13.87 1276.5 4.80 1150.7 14.19 θ6=1340.9 1/s2 1307.8 2.47 1598.7 19.23 1466.1 9.33 1100.6 17.92 θ7=1340.9 1/s2 1441.3 7.49 1336.6 0.32 1668.9 24.46 1392.7 3.86 θ8=1.3409 1/s2 1456.6 8.63 1299.1 3.11 1344.9 0.30 924.4 31.06 θ9=1340.9 1/s2 1346.1 0.38 1317.7 1.73 1065.0 20.57 1144.0 14.68 ω1=11.07 rad/s 10.90 1.51 10.86 1.88 10.81 2.32 11.36 2.66 ω2=31.80 rad/s 31.64 0.49 31.87 0.25 31.66 0.42 31.07 2.27 ω3=48.86 rad/s 48.46 0.81 49.15 0.59 47.51 2.76 48.21 1.33 ω4=61.63 rad/s 62.13 0.81 60.80 1.35 62.77 1.85 62.32 1.13 ω5=70.14 rad/s 70.14 0 71.93 2.55 72.37 3.18 72.75 3.72 ζ1=1 % 1 0 1 0 1 0 0.99 1 ζ2=1 % 1 0 1 0 1 0 0.99 1 ζ3=1.31 % 1.3 0.76 1.31 0 1.28 2.29 1.3 0.76 ζ4=1.57 % 1.58 0.64 1.55 1.27 1.6 1.91 1.59 1.27 ζ5=1.75 % 1.75 0 1.79 2.29 1.8 2.86 1.81 3.43 m1=10.78 Kg 10.78 0 10.78 0 10.78 0 10.78 0 m2=9.20 Kg 9.27 0.77 9.44 2.57 8.27 10.09 15.70 70.62 m3=9.20 Kg 9.86 7.20 8.30 9.77 9.75 5.95 16.78 82.43 m4=9.20 Kg 8.95 2.73 9.93 7.93 8.56 6.92 13.26 44.16 m5=9.20 Kg 9.68 5.26 9.79 6.42 10.81 17.53 10.72 16.48 k1=18415 N/m 17766 3.53 17560 4.64 17731 3.72 31699 72.14 k2=12336 N/m 12229 0.87 11917 3.39 12093 1.97 17335 40.53 k3=12336 N/m 12402 0.87 12675 2.75 12443 0.87 19312 56.55 k4=12336 N/m 12898 4.56 13272 7.57 14291 15.85 18471 49.73 k5=12336 N/m 13035 5.66 12900 4.57 11516 6.65 12260 0.62


### 7.7. Summary of Experimental Results and Discussion

By comparing the results shown in Sections7.1–7.6, the following points are concluded:(i)
The termf∗=1/δ∗ tends to decrease as the number of instrumented floors is reduced. Based on this fact and in our experience, the parameter δ should be selected so that ωc=2π/δ takes a value within the interval ωc∈1.8ωmaxP/n,2.8ωmaxP/n, where ωmax is the maximum natural frequency of the structure.(ii)
Parameterf∘ does not depend on the number of instrumented floors. During experiments, term f∘ 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(1) the cubic spline shape function yields good results in the reconstruction of the unknown responses and (2) 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 ratioP/n is around 0.4–0.6, good results in terms of the performance indexes Eω~(δ) and Ex~(δ) and the reproduction of displacements and velocities can be obtained. Values of P/n 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 P/n 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 P/n=0.5 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 P/n=0.4, corresponding to the proposed method, is larger than the ratios P/n 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.


## 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. Table2 shows the values of Eω~∗, δ∗, Ex~∘, and δ∘ for the unique distribution of five instrumented floors, which is named Π15. Moreover, Figures 4(a) and 4(b) show, respectively, the variation of the performance indexes Eω~(δ) and Ex~(δ), where δ∈[0.01,0.12]. Note that δ∗≠δ∘, indicating that the estimated model with the best quality does not produce the smallest value of Ex~(δ). The reason is that term εa 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 Ex~∘. From Figures 4(a) and 4(b), it is possible to see that Eω~(δ) varies within [1.96,29.47], whereas function Ex~(δ) takes values within the interval [6.37,14.84].Table 2
Results obtained when all floors are instrumented.
Πj5 Instrumented floors Non-instrumented floors E- Eω~∗ δ ∗ [s] f c ∗ [Hz] Ex~∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 2 3 4 5 – – 1.96 0.035 28.57 6.37 0.055 18.18Figure 4
FunctionsEω~(δ) and Ex~(δ) computed when all the floors are instrumented.
(a)
E ω ~ δ versus  δ (b)
E x ~ ( δ ) versus δ


## 7.2. Results with Four Instrumented Floors

Table3 presents the parameters E¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘, which are computed using the distributions Πj4, j=1,2,…,5. Furthermore, Figures 5(a) and 5(b) depict, respectively, the functions Eω~ and Ex~ with respect to δ. From these figures, it is possible to observe that values of Eω~(δ), produced by distributions Πk4, j=1,2,…,4, for δ∈[0.02,0.05], are similar. Moreover, functions Ex~(δ) corresponding to these distributions have a small variation for δ∈[0.05,0.1]. On the other hand, the smallest value of E¯ is obtained with distribution Π34, but this layout does not produce the smallest values of Eω~∗ and Ex~∘. Note that the largest values of Eω~(δ) and Ex~(δ) for δ∈[0.02,0.06] are computed with distribution Π54, i.e., where the top floor is not instrumented.Table 3
Results with four instrumented floors.
Π j 4, j Instrumented floors Non-instrumented floors E- Eω~∗ δ ∗ [s] f c ∗ [Hz] Ex~∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 2 4 5 3 0.18 3.50 0.04 25 11.64 0.048 20.83 2 0 1 3 4 5 2 0.27 3.59 0.038 26.32 9.45 0.058 17.24 3 0 1 2 3 5 4 0.16 4.18 0.04 25 13.38 0.048 20.83 4 0 2 3 4 5 1 0.45 5.79 0.044 22.73 7.50 0.062 16.13 5 0 1 2 3 4 5 0.32 12.32 0.068 14.71 26.07 0.048 20.83Figure 5
FunctionsEω~(δ) and Ex~(δ) computed with four instrumented floors.
(a)
E ω ~ δ versus  δ (b)
E x ~ ( δ ) versus δ


## 7.3. Results with Three Instrumented Floors

Table4 presents the values of E¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘ for distributions Πj3, j=1,2,…,9, of three instrumented floors. It is not possible to identify the building model using layout Π103, since the reconstructed absolute acceleration types at noninstrumented floors produce a regressor vector ϕ^ such that covariance matrices P in (34) and (35) do not exist. On the other side, Figures 6 and 7 show, respectively, the functions Eω~ and Ex~ with respect to δ. Note that distributions Πj3, j=1,2,3, which are uniform or approximately uniform along the building height, allow obtaining small values of Eω~∗ and Ex~∘, which are close to the ones computed with layouts Πj4, j=1,2,3, of four instrumented floors. On the other hand, distributions Πk3, k=7,8,9, that are not uniform over the building height produce the largest values of E¯ and Eω~∗; however, a small value of Ex~∘ is computed with Π93.Table 4
Results with three instrumented floors.
Πj3,j Instrumented floors Non-instrumented floors E - E ω ~ ∗ δ ∗ [s] f c ∗ [Hz] E x ~ ∘ δ ∘ [s] f c ∘ [Hz] 1 0 1 3 5 2 4 0.48 5.17 0.046 21.74 13.88 0.054 18.52 2 0 2 4 5 1 3 0.68 5.21 0.048 20.83 11.98 0.05 20 3 0 2 3 5 1 4 0.63 6.08 0.046 21.74 14.63 0.056 17.86 4 0 1 2 5 3 4 0.53 8.55 0.054 18.52 25.4 0.048 20.83 5 0 1 3 4 2 5 0.61 9.03 0.05 20 26.27 0.046 21.74 6 0 1 2 4 3 5 0.55 9.98 0.05 20 21.83 0.044 22.73 7 0 1 4 5 2 3 0.74 10.98 0.052 19.23 18.71 0.038 26.32 8 0 2 3 4 1 5 0.78 13.15 0.07 14.29 25.88 0.05 20 9 0 3 4 5 1 2 0.98 14.57 0.06 16.67 12.71 0.064 15.63 10 0 1 2 3 4 5 0.96 – – – – – –Figure 6
FunctionEω~(δ) calculated with three instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figure 7
FunctionEx~(δ) computed with three instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figure8 depicts the absolute acceleration types x¨^a2 and x¨^a4 and the two noninstrumented floors, which are reconstructed by the cubic shape function in the configuration Π13. It is shown that these responses are close to their nominal value. On the other side, Figure 9 shows the entries θ^i, i=1,2,…,9, of the estimated vector θ^ provided by the RLS and corresponding to layout Π13. Note that parameters θ^i, i=1,2,…,9, converge to a neighborhood around a constant value in approximately 5 s.Figure 8
Reconstructed signalsx¨^2a and x¨^4a corresponding to the distribution Π13.
(a)
Comparison betweenx¨2a and x¨^2a (b)
Comparison betweenx¨4a and x¨^4aFigure 9
Estimated vectorθ^ produced by the RLS and corresponding to the configuration Π13.
(a)
θ ^ i, i=1,2,3,4,5 (b)
θ ^ i, i=6,7,8,9


## 7.4. Results with Two Instrumented Floors

Values ofE¯, Eω~∗, δ∗, fc∗, Ex~∘, δ∘, and fc∘ for distributions Πj2, j=1,2,…,7, are shown in Table 5. Furthermore, Figures 10 and 11 present, respectively, the performance indexes Eω~ and Ex~ computed by varying δ. Layout Π12, with which the building is instrumented at approximately regular intervals, originates the smallest value of Eω~∗ and small values of E¯ and Ex~∘ among the ones shown in Table 5. On the other hand, distributions Π52 and Π62 that also are approximately uniform along the building height produce large values of Eω~∗ but small values of E¯ and Ex~∘. It is worth mentioning that configurations Πj2, j=7,8,9,10, which are not uniform, generate large values of E¯. Note also that value of Eω~∗, computed with the identified building model corresponding to Π72, is large. Moreover, it is not possible to identify the model of the structure through layouts Πj2, j=8,9,10.Table 5
Results with two instrumented floors.
Π j 2 j Instrumented floors Non-instrumented floors E - E ω ~ ∗ δ ∗ [s] f c ∗ [Hz] E x ~ ∘ δ ∘ [s] f c ∘ [Hz] 1 0 2 4 1 3 5 1.01 5.40 0.054 18.52 20.7 0.042 23.81 2 0 1 4 2 3 5 1.14 12.46 0.074 13.52 28.42 0.048 20.83 3 0 1 5 2 3 4 1.23 14.68 0.084 11.90 29.29 0.04 25 4 0 3 4 1 2 5 1.28 18.35 0.06 16.67 25.37 0.06 16.67 5 0 2 5 1 3 4 0.99 19.05 0.058 17.24 23.83 0.046 21.74 6 0 3 5 1 2 4 1.14 33.9 0.064 15.63 16.12 0.048 20.83 7 0 4 5 1 2 3 1.47 38.54 0.06 16.67 22.71 0.05 20 8 0 2 3 1 4 5 1.41 – – - – – – 9 0 1 3 2 4 5 1.39 – – - – – – 10 0 1 2 3 4 5 2.24 – – – – – –Figure 10
FunctionEω~(δ) computed with two instrumented floors.
(a)
Π k 2, k=1,2,3,4,5 (b)
Π k 2, k=6,7Figure 11
FunctionEx~(δ) computed with two instrumented floors.
(a)
Π k 3, k=1,2,3,4,5 (b)
Π k 3, k=6,7,8,9Figures12(a) and 12(b) present the displacements x^5 estimated by the proposed state observer using layouts Π14 and Π12, respectively. Distribution Π14 was included for comparison purposes to show a degradation in the quality of the displacement reconstruction when using Π12. In addition, Figure 13 compares the velocities x˙^5 obtained using these same distributions. Note that the quality of the reconstruction of x˙^5 is even worse than that of x^5 when using Π12.Figure 12
Estimates produced by the state observer in the distributionsΠ14 and Π12.
(a)
State estimatex^5 corresponding to the distribution Π14 (b)
State estimatex^5 corresponding to the distribution Π12Figure 13
Estimates produced by the state observer in the distributionsΠ14 and Π12.
(a)
State estimatex˙^5 corresponding to the distribution Π14 (b)
State estimatex˙^5 corresponding to the distribution Π12Note that in the distributionΠ14 only the acceleration x¨3a is estimated by means of the cubic spline shape function, whereas in the distribution Π12 three acceleration types x¨1a, x¨3a, and x¨5a are reconstructed. Figures 14(a) and 14(b) depict the acceleration x¨^3a obtained with the distributions Π14 and Π12, respectively. It is shown that both estimates are close to the nominal acceleration x¨3a. On the other hand, Figure 15 presents the other two estimated acceleration types x¨^1a and x¨^5a corresponding to the distribution Π12. By comparing Figures 14(b), 15(a), and 15(b), it is concluded that the acceleration that is best reconstructed in the distribution Π12 is x¨3a.Figure 14
Accelerationx¨^3a obtained with the distributions Π14 and Π12.
(a)
Accelerationx¨^3a corresponding to the distribution Π14 (b)
Accelerationx¨^3a corresponding to the distribution Π12Figure 15
Accelerationx¨^1a and acceleration x¨^5a corresponding to the distributions Π14 and Π12.
(a)
Accelerationx¨^1a obtained with the distribution Π12 (b)
Accelerationx¨^5a obtained with the distribution Π12


## 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 vectorϕ^, with which the covariance matrices P in (34) and (35) do not exist.


## 7.6. Parameter Estimates Obtained with Two, Three, Four, and Five Instrumented Floors

Define(ΠjP,δ∗) as the jth distribution of P instrumented floors with its corresponding integration period δ∗ of the LIF that produces Eω~∗. Table 6 presents the estimated parameters θ^i, i=1,2,…,9, by the proposed technique with (Π12,δ∗), (Π13,δ∗), (Π14,δ∗), and (Π15,δ∗). Moreover, this Table shows parameters ω^r, ζ^r, m^r and k^r, r=1,2,…,5 computed using those layouts. It is possible to observe that the error in all the estimated natural frequencies ω^r and damping factors ζ^r is less than 4%. Note that for (Π15,δ∗), (Π14,δ∗), (Π13,δ∗), and (Π12,δ∗), the error of the parameter estimates θ^i, i=1,2,…,9, is less than 10%, 20%, 25%, and 73%, 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 (Π13,δ∗) and (Π14,δ∗) have an error less than 18% and 8%, respectively.Table 6
Parameter estimates corresponding to the distributionsΠ12, Π13, Π14, and Π15 of two, three, four, and five floors.
Nominal value Unit Estimate(Π15,δ∗) Error% Estimates(Π14,δ∗) Error% Estimates(Π13,δ∗) Error% Estimates(Π12,δ∗) Error% θ 1 = 1708.3 1/s2 1648.0 3.53 1629.0 4.64 1644.8 3.72 2940.5 72.13 θ2=1144.3 1/s2 1134.4 0.86 1105.5 3.39 1121.8 1.96 1608.1 40.53 θ3=1340.9 1/s2 1319.1 1.63 1262.9 5.82 1462.1 9.04 1104.4 17.64 θ4=1340.9 1/s2 1337.7 0.24 1343.2 0.17 1504.3 12.19 1230.3 8.25 θ5=1340.9 1/s2 1257.5 6.22 1526.8 13.87 1276.5 4.80 1150.7 14.19 θ6=1340.9 1/s2 1307.8 2.47 1598.7 19.23 1466.1 9.33 1100.6 17.92 θ7=1340.9 1/s2 1441.3 7.49 1336.6 0.32 1668.9 24.46 1392.7 3.86 θ8=1.3409 1/s2 1456.6 8.63 1299.1 3.11 1344.9 0.30 924.4 31.06 θ9=1340.9 1/s2 1346.1 0.38 1317.7 1.73 1065.0 20.57 1144.0 14.68 ω1=11.07 rad/s 10.90 1.51 10.86 1.88 10.81 2.32 11.36 2.66 ω2=31.80 rad/s 31.64 0.49 31.87 0.25 31.66 0.42 31.07 2.27 ω3=48.86 rad/s 48.46 0.81 49.15 0.59 47.51 2.76 48.21 1.33 ω4=61.63 rad/s 62.13 0.81 60.80 1.35 62.77 1.85 62.32 1.13 ω5=70.14 rad/s 70.14 0 71.93 2.55 72.37 3.18 72.75 3.72 ζ1=1 % 1 0 1 0 1 0 0.99 1 ζ2=1 % 1 0 1 0 1 0 0.99 1 ζ3=1.31 % 1.3 0.76 1.31 0 1.28 2.29 1.3 0.76 ζ4=1.57 % 1.58 0.64 1.55 1.27 1.6 1.91 1.59 1.27 ζ5=1.75 % 1.75 0 1.79 2.29 1.8 2.86 1.81 3.43 m1=10.78 Kg 10.78 0 10.78 0 10.78 0 10.78 0 m2=9.20 Kg 9.27 0.77 9.44 2.57 8.27 10.09 15.70 70.62 m3=9.20 Kg 9.86 7.20 8.30 9.77 9.75 5.95 16.78 82.43 m4=9.20 Kg 8.95 2.73 9.93 7.93 8.56 6.92 13.26 44.16 m5=9.20 Kg 9.68 5.26 9.79 6.42 10.81 17.53 10.72 16.48 k1=18415 N/m 17766 3.53 17560 4.64 17731 3.72 31699 72.14 k2=12336 N/m 12229 0.87 11917 3.39 12093 1.97 17335 40.53 k3=12336 N/m 12402 0.87 12675 2.75 12443 0.87 19312 56.55 k4=12336 N/m 12898 4.56 13272 7.57 14291 15.85 18471 49.73 k5=12336 N/m 13035 5.66 12900 4.57 11516 6.65 12260 0.62


## 7.7. Summary of Experimental Results and Discussion

By comparing the results shown in Sections7.1–7.6, the following points are concluded:(i)
The termf∗=1/δ∗ tends to decrease as the number of instrumented floors is reduced. Based on this fact and in our experience, the parameter δ should be selected so that ωc=2π/δ takes a value within the interval ωc∈1.8ωmaxP/n,2.8ωmaxP/n, where ωmax is the maximum natural frequency of the structure.(ii)
Parameterf∘ does not depend on the number of instrumented floors. During experiments, term f∘ 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(1) the cubic spline shape function yields good results in the reconstruction of the unknown responses and (2) 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 ratioP/n is around 0.4–0.6, good results in terms of the performance indexes Eω~(δ) and Ex~(δ) and the reproduction of displacements and velocities can be obtained. Values of P/n 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 P/n 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 P/n=0.5 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 P/n=0.4, corresponding to the proposed method, is larger than the ratios P/n 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:(1) it is not possible to estimate the model and state of the structure with only one instrumented floor; (2) 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; (3) increasing the number of instrumented floors improves the quality of the estimated model and reduces the state estimation error; (4) 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; (5) 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.



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