Output-only modal identification by compressed sensing: Non-uniform low-rate random sampling

Mechanical Systems and Signal Processing 56-57 (2015) 15–34

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Output-only modal identification by compressed sensing: Non-uniform low-rate random sampling Yongchao Yang a, Satish Nagarajaiah a,b,n a b

Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, USA Department of Mechanical Engineering, Rice University, Houston, TX 77005, USA

a r t i c l e i n f o

abstract

Article history: Received 5 October 2013 Received in revised form 22 September 2014 Accepted 29 October 2014 Available online 20 November 2014

Modal identification or testing of structures consists of two phases, namely, data acquisition and data analysis. Some structures, such as aircrafts, high-speed machines, and plate-like civil structures, have active modes in the high-frequency range when subjected to high-speed or broadband excitation in their operational conditions. In the data acquisition stage, the Shannon–Nyquist sampling theorem indicates that capturing the high-frequency modes (signals) requires uniform high-rate sampling, resulting in sensing too many samples, which potentially impose burdens on the data transfer (especially in wireless platform) and data analysis stage. This paper explores a new-emerging, alternative, signal sampling and analysis technique, compressed sensing, and investigates the feasibility of a new method for output-only modal identification of structures in a non-uniform low-rate random sensing framework based on a combination of compressed sensing (CS) and blind source separation (BSS). Specifically, in the data acquisition stage, CS sensors sample few nonuniform low-rate random measurements of the structural responses signals, which turn out to be sufficient to capture the underlying mode information. Then in the data analysis stage, the proposed method uses the BSS technique, complexity pursuit (CP) recently explored by the authors, to directly decouple the non-uniform low-rate random samples of the structural responses, simultaneously yielding the mode shape matrix as well as the non-uniform low-rate random samples of the modal responses. Finally, CS with ℓ1 -minimization recovers the uniform high-rate modal response from the CP-decoupled nonuniform low-rate random samples of the modal response, thereby enabling estimation of the frequency and damping ratio. Because CS sensors are currently in laboratory prototypes and not yet commercially available, their functionality—randomly sensing few non-uniform samples—is simulated in this study, which is performed on the examples of a numerical structural model, an experimental bench-scale structural model, and a realworld seismic-excited base-isolated hospital buildings. Results show that the proposed method in the CS framework can identify the modes using non-uniform low-rate random sensing, which is far below what is required by the Nyquist sampling theorem. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Output-only modal identification Compressed sensing Low-rate random sampling Blind source separation Complexity pursuit Seismic responses

n Corresponding author at: Department of Civil and Environmental Engineering, Rice University, Houston, TX 77005, USA. Tel.: þ 1 713 348 6207; fax: þ1 713 348 5268. E-mail addresses: [email protected] (Y. Yang), [email protected] (S. Nagarajaiah).

http://dx.doi.org/10.1016/j.ymssp.2014.10.015 0888-3270/& 2014 Elsevier Ltd. All rights reserved.

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1. Introduction 1.1. Motivation: modal identification in traditional Nyquist sampling paradigm Experimental modal identification of structures conventionally relies on the measurement of the frequency response function (FRF), which is based on the relationship between system inputs and outputs. This corresponds to modal testing where excitation can be controlled and accurate measurement of inputs and outputs is possible [1]. Operational or outputonly modal identification [2], on the other hand, provides a practical alternative when excitation information (e.g., wind, traffic, etc) is extremely difficult to obtain accurately. Modal identification, whether experimental or operational, consists of two phases, namely, data acquisition and data analysis. Some structures (e.g., aircrafts, high-speed machines, plate-like civil structures, etc) have active modes of interest in the high frequency range, when confronting high-speed or broadband excitation in their operational conditions. In such a case, the information of these active modes in high-frequency range is essential for structural dynamic analysis and model updating [3]. Capturing these active, potentially high-frequency, modes, in the traditional Shannon–Nyquist paradigm following the celebrated sampling theorem, requires extremely high uniform sampling rate (at least twice the maximum frequency of the highest active mode) in the data acquisition phase. Although feasible, it would result in sensing a large number of samples, which could significantly increase the computational demand in the data analysis (modal parameter estimation) phase. Especially, when the data acquisition is performed in the wireless sensor platform with only limited power and communication resources [4], transferring these resultant large number of samples to the data analysis base station would be a heavy burden. 1.2. Compressed sensing: an alternative sensing technique Attempting to address these concerns, this paper explores an alternative, new-emerging signal sampling technique, called compressed sensing (CS) [5–7], and investigates the feasibility of performing output-only modal identification of structures in the CS framework—non-uniform low-rate random sampling. Different from the traditional Nyquist uniform sampling wisdom, CS enables non-uniform low-rate random sampling (Fig. 1 for a simple example) of a signal to capture sufficient information for exact recovery. When the underlying signal is sparse in a proper representation domain, CS is able to exactly recover it from far fewer incoherent random measurements than what is required by the sampling theorem. Intuitively, the two key ingredients—sparsity and incoherence—are satisfied by the output-only modal identification problem: the targeted monotone modal responses are spectrally sparsest in the frequency domain, which is maximally incoherent with the non-uniform random sensing space (time domain). In the proposed method, CS is used in the data acquisition stage, sampling few low-rate random measurements of the structural responses signals. 1.3. Output-only modal identification in compressed sensing framework In the data analysis stage, the blind source separation (BSS) [8] technique is cast to this novel low-rate random sensing framework to perform output-only modal identification using only the available non-uniform low-rate random measurements of the structural responses signals. As an unsupervised learning algorithm, BSS is able to recover the unknown sources using only the mixture data set based on the general assumption that sources are statistically independent. A family of BSS techniques, independent component analysis (ICA) [9], second order blind identification (SOBI) [10], and complexity pursuit (CP) [11,12], have been successfully used in output-only modal identification [13–21]. While recent research in the signal processing community has also explored BSS using the CS principle [22–26], this study focuses on exploiting the BSS assumption of independence of sources (viewed as modal responses), which is found to enable direct modal separation of the few low-rate random samples of the structural responses. Specifically, the CP method recently explored by the authors [21] is used in the proposed method; its performance is also compared with other BSS based methods. The proposed method combines CS and CP in a non-uniform low-rate random sensing framework. In the signal acquisition stage, the non-uniform spike samplers randomly sense few samples of the structural responses at a significantly low rate, while retaining the underlying modal information of the structure. Then in the data analysis stage, the CP algorithm directly decouples the measured non-uniform low-rate random samples of the structural responses, simultaneously yielding the mode shapes, as well as the decoupled non-uniform low-rate random samples of the estimated modal response. From the low-rate random samples of each modal response, the ℓ1 -minimization recovers the uniform high-rate modal response, thereby enabling estimation of the frequency and damping ratio. 1.4. Potentials of the CS framework in modal identification Compared to those within traditional Nyquist sampling paradigm, the advantage of the proposed method in CS framework is that it avoids sampling too many samples in the data acquisition step, but still is able to capture the active modes potentially in high-frequency range, saving many resources (especially in wireless sensor platform) in transferring and analyzing a large number of samples, which would otherwise be required by traditional Nyquist sampling paradigm. In

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Fig. 1. The distinction between traditional Nyquist sampling and compressed sensing of a time domain signal. (a) Nyquist sampling needs to sense N samples of a signal (at an uniform rate at least twice the maximum signal frequency) to avoid information loss; and (b) compressed sensing randomly captures much fewer M⪡N non-uniform low-rate samples of a spectrally sparse signal, which is sufficient for exact recovery (later by ℓ1 -minimization). Here shows a small segment (10–15 s) of the system responses (displacements) of the numerical model (Fig. 3) for visual enhancement.

addition, recent study [27–30] has shown that the CS based samplers (in laboratory prototype) are more resource-efficient (e.g., power) than traditional Nyquist samplers in many applications, especially in sampling the high-rate spectrally sparse signals. Lately, the CS technique has been explored by researchers in structural health monitoring and system identification, showing encouraging results [31–33].

1.5. Assumption of this study Although the prototype of the non-uniform low-rate CS based sampler has been designed very lately [27–30], as well as vast novel CS hardware instruments [34,35], currently, there are no CS sensors commercially available. Therefore, in this study, the functionality of CS samplers is simulated to validate the proposed CP–CS method, where they randomly sample few non-uniform measurements of the structural responses of a numerical model, an experimental bench-scale structural model, and a real-world seismic-excited base-isolated hospital building. Results show that the proposed method can capture the modes using far fewer low-rate random samples (or lower sampling rate) than what is required by the Nyquist sampling theorem; they suggest that the proposed method using non-uniform low-rate random sensing can serve as an efficient alternative to traditional Nyquist sampling in modal identification, once the CS samplers become commercially available in the near future. This paper is organized as follows: Section 2 introduces the theory of output-only modal identification based on the BSS technique CP. Section 3 focuses on the CS theory and Section 4 on the proposed CP–CS method. Section 5 summarizes the procedures of the proposed CP–CS method. Simulation study is performed on a numerical model in Section 6 and on anexperimental bench-scale structural model as well as a real-world seismic-excited base-isolated hospital building. Section 8 concludes the paper and discusses future study.

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2. Blind source separation and output-only modal identification 2.1. Blind source separation and modal expansion Blind source separation (BSS) is an unsupervised learning technique which recovers the characteristics hidden in the multivariate data set. BSS assumes the m measured mixture signals x(t) ¼[x1(t),...,xm(t)]T are generated by a linear instantaneous mixing model [9] n

xðtÞ ¼ AsðtÞ ¼ ∑ ai si ðtÞ

ð1Þ

i¼1

where A is the unknown m  n mixing matrix and s(t) ¼[s1(t)...,sn(t)]T are the n latent sources, both to be estimated simultaneously (hence the term “blind”); ai A ℝm is the ith column of A associated with si (row vector), containing its characteristics of the mixing process. The square BSS model is adopted in this study, i.e., m¼ n and ai A ℝm . When m 4n, principal component analysis (PCA) can be used to reduce the dimensionality to a square BSS model, and the underdetermined BSS model with m on is beyond the scope of this study. With only x(t) known, the BSS problem cannot be solved mathematically. Additional assumption is thus needed to estimate the BSS model. Most BSS techniques, such as independent component analysis (ICA), second order blind identification (SOBI), and complexity pursuit (CP), make the assumption that the sources s(t) are statistically independent (SOBI assumes uncorrelatedness) at each time instant t; i.e., any two sources si(t) and sj(t) (i,j¼1,...,n,i aj) are independent at each t. Generally, it suffices to estimate both the sources and mixing matrix in most applications [9]. The BSS model has a remarkable resemblance with the modal expansion of the system responses x(t) (measured) as a linear combination of n modal responses (to be identified) q(t) ¼ [q1(t),...,qn(t)]T n

xðtÞ ¼ ΦqðtÞ ¼ ∑ φi qi ðtÞ

ð2Þ

i¼1

from an n degree-of-freedom (DOF) linear system with an equation of motion (EOM) € þ CxðtÞ _ þ KxðtÞ ¼ fðtÞ MxðtÞ

ð3Þ

where M, C, and K are constant mass, damping, and stiffness matrices, respectively, and are assumed to be real-valued and symmetric; Φ is the vibration modeshape matrix (to be identified), whose ith column φi (modeshape) is associated with qi (row vector). Kerschen et al. [14] proposed that if the modal responses have incommensurable frequencies, then q(t) can be viewed as independent sources, and considering x(t) in the modal expansion (Eq. (2)) as mixtures in the BSS model (Eq. (1)) would lead to the recovery of s(t) and A, which correspond to q(t) and Φ, respectively. ICA and SOBI have been applied to perform output-only modal identification and found to be efficient and straightforward. However, ICA is restricted to lightly damped structures [14,18], while it is difficult to identify closely-spaced modes and highly non-diagonalizable modes using SOBI [16,17]. Modified ICA [18] and SOBI [16,17,20] have been proposed to overcome some of these limitations. Recently, the authors have explored the CP method (in the traditional Nyquist sampling) and found it to be a useful alternative for identification of closely-spaced modes coupled with high damping [21]; it is used in the proposed method and reviewed briefly in the following, while its performance (as well as in non-uniform low-rate random sensing) is compared to ICA and SOBI in Section 6.2.2. 2.2. Complexity pursuit for output-only modal identification To solve the BSS problem, the theoretically-justified Stone's theorem [36] states that the complexity of any mixture is always higher than the simplest constituent source. Using this basis to implement the CP learning rule, the simplest signal extracted from a set of mixtures would be a source signal. Relating to the modal expansion where any system response xi(t) (i¼1,...,n) is a linear mixture of the modal responses s(t), the complexity of any xi(t) (i¼ 1,...,n) will always be higher than that of the simplest modal response. Specifically, CP seeks a de-mixing (row) vector wi A ℝn such that the recovered component yi(t) yi ðtÞ ¼ wi xðtÞ

ð4Þ

yields the least complexity and thus approaches the simplest source signal. The complexity of a signal is measured by temporal predictability [11]
 Vðyi Þ ∑t ðy i ðtÞ  yi ðtÞÞ2 ¼ log F yi ¼ log 2 Uðyi Þ ∑t ðy^ i ðtÞ  yi ðtÞÞ

ð5Þ

in which the long-term predictor y i ðtÞ and the short-term predictor y^ i ðtÞ are defined by y i ðtÞ ¼ λL y i ðt  1Þ þð1  λL Þyi ðt 1Þ

0 r λL r1

y^ i ðtÞ ¼ λS y^ i ðt  1Þ þð1  λS Þyi ðt  1Þ

0 r λS r 1

ð6Þ

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where λ ¼2  1/h with hS ¼1 and hL is arbitrarily set (say, hL ¼900,000, which is used in all the examples of this study) as long as hL⪢hS [11]. While it is possible to extract the simplest source (modal response) one by one—after the simplest source is extracted, it is removed using the Gram–Schmidt orthonormalization—Stone proposed a more efficient algorithm which formulates the BSS model incorporating the contrast function Eq. (5) to a generalized eigenproblem [11,37] by computing the long-term and short-term covariance matrices using the fast convolution operations, thereby simultaneously recovering all the sources y(t)¼[y1(t)...,yn(t)]T yðtÞ ¼ WxðtÞ

ð7Þ

which approximates the source vector s(t). W (the ith row is wi) is the simultaneously recovered de-mixing matrix. Incorporating this into the modal identification framework (relating Eq. (2) and Eq. (1)), where the modal responses are viewed as sources and system response as mixtures, qðtÞ ¼ yðtÞ ¼ WxðtÞ

ð8Þ

and the vibration modeshape matrix can be estimated by

Φ ¼ W1

ð9Þ

The frequency and damping ratio can then be readily computed from the recovered time-domain modal response q(t) using Fourier transform (FT) and the logarithm decrement technique [38], respectively. For details of the CP output-only modal identification method (in traditional Nyquist (uniform) sensing), the readers are referred to Refs. [21,33]. Its applicability, as compared to other BSS techniques, in the non-uniform low-rate random sensing framework, is discussed in Section 4.2 with examples in Section 6.2.2. 3. Compressed sensing Traditional sampling protocol follows the Shannon–Nyquist sampling theorem, which states that the sampling frequency must be at least twice the maximum frequency of the signal to avoid information loss. CS [5–7] alleviates this limitation, stating that it is possible to exactly recover sparse signals from far fewer low-rate incoherent random samples than what is required by the sampling theorem. The ingredients of CS are sparsity, incoherence, and randomness, which are reviewed briefly in this section. 3.1. Under-sensing by CS CS [7] performs linear sensing of an N-dimensional signal R A ℝNM (row vector) by correlating it with the sensing waveform yi A ℝM (row vector), e.g., spikes, sinusoids, wavelets, etc, bk ¼ 〈f ; υk 〉;

k ¼ 1; :::; M

ð10Þ

yielding only M (M⪡N, highly under-sampling) observations b A ℝ operator. Using a compact matrix denotation,

M

b¼fϒ

(bk is the kth sample); 〈  〉 denotes the inner product ð11Þ

where ϒ ¼ ½υ1 ; :::; υM  A ℝ is the sensing matrix whose kth column is υk. For example, when the sensing waveform υk (k ¼1,...,M) is the Dirac delta function or spike function (used in this study), it samples the signal in the original time domain: it reduces to the traditional Nyquist (uniform) sampling (Fig. 1(a)) if M¼N; as opposed, M⪡N in CS (Fig. 1(b)). In CS, because only M⪡N samples are available, recovery of the underlying f A ℝN from the incomplete knowledge of the few observations b A ℝM and ϒ A ℝNM is generally ill-posed: there exist infinite solutions. CS offers a successful solution by exploiting the sparsity of the underlying signal and skillfully designing the sensing modality—incoherent random sensing, as introduced in the following: NM

3.2. Incoherent random sensing and sparsity recovery The signal f A ℝN can be represented in a proper domain, N

f ¼ αΨ ¼ ∑ αj ψj j¼1

ð12Þ

where Ψ ¼ ½ψ1 ; :::; ψN T A ℝNN is an orthonormal basis (e.g., sinusoid, wavelet, etc), whose jth row is ψj A ℝN (or ℂN on Fourier basis). α A ℝN (row vector) is the coefficient sequence of f A ℝN on Ψ, whose jth element αj ¼ 〈f,ψj〉. α is said to be K-sparse if at most K elements in α are non-zero; in this case, f is also understood to be K-sparse. A simple example is the sinusoid, which is sparsest (K ¼1) in the frequency domain.

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CS requires small coherence between the representation space Ψ ¼ ½ψ1 ; :::; ψN T A ℝNN and the sensing space ϒ ¼ ½υ1 ; :::; υN  A ℝNN , defined by pffiffiffiffi   μϒ; Ψ ¼ N max〈υj ; ψl 〉; 1 r j; l r N ð13Þ pffiffiffiffi Coherence measures the correlation between two bases with a bound of [1; N]. For example, the coherence between the spikes and sinusoids is unit, i.e., they are maximally incoherent. Substituting Eq. (12) into Eq. (11) yields a new form of the under-sampled recovery problem b ¼ αΨϒ ¼ αΘ

ð14Þ

is the new sensing matrix that senses partial information of α. CS states that if α is K-sparse, and M where Θ ¼ Ψϒ A ℝ measurements in ϒ domain are randomly (with a uniform distribution) sensed, then with overwhelming probability, α A ℝN (N⪢M) can be exactly recovered by the convex program ℓ1 -minimization, which is known as basis pursuit [39], NM

ðP 1 Þ:

αn ¼ arg min ‖α‖ℓ1 subject to αΘ ¼ b

ð15Þ

provided that M Z C μ2 ðϒ; ΨÞK log N

 

 j . J ℓ1 ¼ ∑N j¼1

ð16Þ

α Then the underlying f A ℝ can be where C is some small positive constant, and the ℓ1 -norm is defined as J α readily obtained by f ¼ αn Ψ. Eq. (16) says that the sparser the signal (the smaller K) and the larger incoherence ðϒ; ΨÞ (the smaller μ) is, the smaller M would be required. If, for example, ðϒ; ΨÞ are the spikes and the sinusoids with μ ¼1, then randomly sensing only M p K log N samples is sufficient for ℓ1 -minimization to exactly recover α A ℝN . If the representation of f A ℝN , α, is truly sparse in Ψ, i.e., K is indeed small, then M can be much smaller than N or what is required by the Nyquist sampling theorem. Another situation often arises that the signal may be considerably corrupted by noise. Then it is more appropriate to consider a stable version of (P1), termed basis pursuit denoising, ðP ε1 Þ: where

αn ¼ arg min ‖α‖ℓ1 subject to J b αΘ J ℓ2 r ε

N

ð17Þ

ε is the bound associated with the noise level.

4. BSS based output-only modal identification in CS framework As introduced in Section 3, successful application of CS requires signal sparsity on some appropriate domain and incoherence between the signal representation and sensing spaces. The proposed output-only modal identification framework obeys these requirements: (1) the underlying monotone modal response has sparsest representation in the frequency domain; and (2) the non-uniform low-rate random sensing space is maximally incoherent with the frequency domain where the modal response is sparsely represented. As CS does not assume a priori knowledge about the distribution or amplitude of the sparse elements (in a proper domain) present in the sensed signal, its non-uniform low-rate random sampling could be used as an alternative to the uniform Nyquist sensing for sampling the high-frequency signal, because it is spectrally sparse as in a high-frequency mode. This section casts the BSS based output-only modal identification method into the CS low-rate random sensing framework. It is introduced step by step in the following: the data acquisition stage is Step 1 and the data analysis stage (performed at base station) includes Steps 2 and 3. 4.1. Step 1: compressed sensing of the system responses Supposing the n-DOF system Eq. (3) is outputting N-dimensional structural responses X A ℝnN (at the ith channel, xi A ℝN (i¼1,...,n)), then in Step 1 data acquisition stage, the non-uniform low-rate CS sensor randomly senses few M⪡N samples, at each channel, yielding the non-uniform low-rate random measurements X A ℝnM (xi A ℝM ). If mathematically formulated and incorporated into the modal expansion Eq. (2) with the unknown Φ A ℝnn (φi A ℝn ) and Q A ℝnN (qi A ℝN ), it amounts to X ¼ XR ¼ ΦQR ¼ ΦðQRÞ ¼ ∑ni¼ 1 φi ðqi RÞ ¼ ∑ni¼ 1 φi qi

ð18Þ

where R A ℝNM denotes the non-uniform low-rate sampler, whose kth column rk ¼ δðt  τÞ A ℝN (k¼1,...,M) is a spike function and τ A [1,N] is determined by randomly selecting the M sensing locations. Note that in the non-uniform low-rate random sensing framework (Fig. 1(b)), the sampling rate is time-varying or instantaneous; as opposed, the sampling rate is uniform in traditional Nyquist sampling (Fig. 1(a)). Discussion: With the non-uniform low-rate random samples of the system responses X A ℝnM , in the data analysis stage, according to the CS theory (Eq. (16)), it is possible to use ℓ1 -minimization to recover the structural response xi A ℝN (K¼ nsparse in the frequency domain, or equals the active mode number) directly from xi A ℝM first, if (following Eq. (16)) M Z Cn log N

ð19Þ

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since μ ¼1 between the spike sensing space and the frequency representation space. Then output-only modal identification can be performed on the recovered X A ℝnN by any output-only methods such as BSS. However, much smaller M Z C log N can be further achieved by the proposed approach using BSS, as described in the following. 4.2. Step 2: direct decoupling the low-rate random samples of system responses by BSS Eq. (18) indicates that randomly sensing the system responses equals a linear combination of randomly sensing the modal responses on each of the independent modal coordinates. This implies that if X A ℝnM can be first decoupled into Q A ℝnM in Eq. (18), then using ℓ1 -minimization for recovery of each qi A ℝN (i ¼1,...,n) from its low-rate random samples qi A ℝM would require least M Z C log N random measurements; since, the underlying qi A ℝN are spectrally sparsest (K ¼1sparse). The independence assumption of BSS permits its applicability even in the non-uniform low-rate random sensing framework. To see this, rewrite Eq. (18) as the instantaneous model n

xðkÞ ¼ ΦqðkÞ ¼ ∑ φi qi ðkÞ

ð20Þ

i¼1

Because BSS assumes that the sources are statistically independent (SOBI assumes uncorrelatedness) at each time instant t (t¼1,...,N), at any random sensing instant k (k ¼ 1,...,M C[1,N]), the source independence holds; this implies that BSS is equally applicable to a set of low-rate random sensed samples. Also note that using the same CS sampler R in Eq. (18) at all channels ensures that all the xi (i¼1,...,n) are sensed at the same instant k, when their independence holds. Fig. 2 shows the non-uniform low-rate random sampling with a 3-channel example. Following the CP learning rule, as introduced in Section 2.2, even with M⪡N non-uniform low-rate random measurements, CP can perform output-only modal identification (decouple) qðkÞ ¼ yðkÞ ¼ WxðkÞ

ð21Þ

~ ¼ W  1 as the estimation of the modeshape matrix, and simultaneously recovering qðkÞ ¼ yðkÞ with directly yielding Φ M qi ¼ yi A ℝ (i¼ 1,...,n), which is the non-uniform low-rate random samples of the estimated modal response. The following example demonstrates that the CP learning rule for modal identification holds even for low-rate randomly-sensed signals. Suppose there are five N ¼100,000-dimensional source signals, which are zero-mean sinusoids with random variance and different frequencies (Table 1). Each of them is sampled randomly (with a uniform distribution), yielding only M¼1500 non-uniform low-rate random measurements (the average sensing rate is 100 Hz  (1500/100,000) ¼1.5 Hz, much lower than what is required by the Nyquist sampling theorem). The low-rate random-sensed sources are then mixed by a normally-distributed random mixing matrix, resulting in five mixtures with non-uniform low-rate random samples. The complexity of the uniformly sensed sources and mixtures, and that of the low-rate randomly sensed sources and mixtures, are computed using the predictability contrast function Eq. (5). Because of the randomness, this simulation procedure is repeated 100 times. It can be seen that the CP learning rule is valid even for the randomly sensed signals: the mixtures are always more complex than the simplest source. Table 1 presents the result for the 100th run for illustration. Clearly the predictability of the simplest source (its frequency is 0.1 Hz) is larger than any of the mixtures, whether in Nyquist uniform sensing or nonuniform low-rate random sensing. 4.3. Step 3: sparse recovery of modal responses The remaining problem is recovery of qi A ℝN (i¼1,...,n) to estimate the frequency and damping ratio, from the knowledge of the CP-decoupled non-uniform low-rate random samples of the estimated modal response yi ¼ qi ¼ qi R A ℝM and the sensing matrix R A ℝNM . Use the fact that qi A ℝN is monotone in the sinusoid basis, qi ¼ αi C

ð22Þ

vector) is a very sparse representation sequence (ideally K ¼1 sparse) of qi A ℝ , and C ¼[c1,...,cN] whose where αi A ℝ (row pffiffiffiffiffiffiffiffiffi jth row is cj ðtÞ ¼ 2=N cos ðπ ð2t þ 1Þj=2NÞ (t ¼0,...,N  1), it then becomes yi ¼ αi CR A ℝM (i ¼1,...,n). Combining with the maximal incoherence μ(Δ,C)¼ 1, CS exactly recovers αni ¼ αi A ℝN (i¼ 1,...,n) with overwhelming probability by the ℓ1 -minimization, N

ðP 1 Þ:

N

αni ¼ arg min‖αi ‖ℓ1 subject to αi CR ¼ yi

T

ð23Þ

if the M(⪡N) non-uniform low-rate random measurements satisfy (following Eq. (16)) M Z C log N

ð24Þ

which is much smaller than M ZCn log N (Eq. (19)). Then the time-domain modal response qi A ℝ (i¼1,...,n) is obtained by N

qi ¼ αni C

ð25Þ

from which the frequency and damping ratio (in free vibration) are estimated by Fourier transform (FT) and logarithm decrement, respectively.

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Fig. 2. The non-uniform low-rate random sensing paradigm for output-only modal identification using blind source separation (BSS) technique complexity pursuit (CP) and compressed sensing (CS); a 3-channel example is shown with the details described in Section 6 (only the 10–20 s is shown for visual enhancement). In Step 1 (data acquisition), at each of the three channel (i¼ 1,2,3), the non-uniform low-rate random sampler (denoted by the sensing matrix R A ℝNM ) randomly senses M⪡N non-uniform low-rate samples xi A ℝM of the structural response xi A ℝN , yielding the non-uniform low-rate random samples X A ℝ3M . In Step 2 (data analysis), CP blindly and directly decouples the non-uniform low-rate random samples X A ℝ3M by Y ¼ WX, simultaneously yielding the inverse de-mixing matrix as the estimation of the modeshape matrix Φ ¼ W  1 A ℝ33 (each column corresponds to one mode), and the decoupled non-uniform low-rate random samples of the estimated modal responses Q ¼ Y A ℝ3M (qi ¼ yi A ℝM , i¼1,2,3). In Step 3 (data analysis continued), for each mode, from the knowledge of the random sensing matrix R A ℝNM and the CP-decoupled non-uniform low-rate random samples of the estimated modal response yi A ℝM , ℓ1 -minimization (see Eqs. (23)–(25)) recovers the uniform high-rate modal response xi ðtÞ, where frequency can be estimated by Fourier transform and damping ratio by logarithm decrement (in free vibration).

Table 1 The predictability (by CP) of sinusoids and their mixtures (N ¼100,000, M ¼1500). Source

1 2 3 4 5

Frequency

0.1 1 π pffiffiffiffiffiffi 15 10

Predictability

Mixture

Uniform

Random

1.4823 1.3528 1.1053 0.9752

0.5163  0.0755  0.0897  0.0992

0.2306

 0.1062

Predictability Uniform

Random

1 2 3 4

1.0341 1.1154 1.3502 1.0659

 0.0882  0.0962  0.1025  0.1009

5

0.9864

 0.0943

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Note that the choice of the discrete cosine transform (DCT) basis as per Eqs. (22)–(25) can avoid complex elements in solving (P1), compared to the standard discrete Fourier transform (DFT) basis. This study adopts the efficient ℓ1 -regularized least squares algorithm [40] to solve (P1) with the regularization parameter set at λ ¼0.01 (the solution is not sensitive to a large range of the regularization parameter λ). The advantage of performing CP directly on the non-uniform low-rate random samples of the system responses X A ℝnM becomes clear: (1) one correctly obtains the mode shapes directly from the seemingly incomplete X A ℝnM ; (2) the required measurement M⪡N is significantly reduced to M Z C log N (K ¼1-sparse), comparing to M Z Cn log N (K ¼n-sparse) (Eq. (19)) if one chooses to first recover X A ℝnN from X A ℝnM , and then perform output-only modal identification on X A ℝnN . Their performances are compared in Section 6.2.4. 5. The procedures of CP–CS output-only modal identification method The proposed CP–CS output-only modal identification method using low-rate random sensing follows several steps to implement (Fig. 2 for illustrations): Data acquisition: Step 1. Use the non-uniform low-rate sensors to sample the structural responses randomly. Data analysis: Step 2. Perform CP directly on the non-uniform low-rate random samples of the structural responses, simultaneously obtaining the mode shape matrix and the non-uniform low-rate random samples of the estimated modal responses; Step 3. From the CP-decoupled non-uniform low-rate random samples of the estimated modal response, use ℓ1 -minimization to recover the uniform high-rate modal response, whereby estimating the frequency (Fourier transform) and damping ratio (logarithm decrement in free vibration). 6. Numerical simulations 6.1. Numerical model To demonstrate the performance of the proposed modal identification method in the CS framework, numerical simulations are conducted on a 3-DOF linear time-invariant mass-spring damped model (Fig. 3(a)), which is an example from Ref. [14]. The parameters are the same as in Ref. [14]: the lumped mass m1 ¼2, m2 ¼1, m3 ¼3, the spring stiffness k1 ¼k12 ¼k23 ¼k3 ¼1, and proportional damping is considered with respect to the mass matrix, C ¼ βM, with β ¼0.01,0.05,0.10 to account for different damping levels. Both free vibration and random vibration are studied. Newmark-Beta algorithm is used to integrate the EOM of the system, which is outputting N-dimensional time histories of the structural responses X A ℝ3N (xi A ℝN ,i ¼1,2,3). Step 1 data acquisition: at each channel, the CS sensor randomly senses M⪡N non-uniform low-rate samples X A ℝ3M (xi A ℝM , i¼1,...,3, M⪡N). The data analysis steps are then conducted. Step 2: CP performs (Step 2 of Fig. 2) output-only modal identification directly on X A ℝ3M , simultaneously yielding the non-uniform low-rate random samples of the estimated ~ , whose accuracy at the ith modal responses, Q A ℝ3M (qi A ℝM ,i ¼1,2,3), as well as the estimation of the mode matrix, Φ (i¼1,2,3) mode is evaluated by the modal assurance criterion (MAC), ~ i ; φi Þ ¼ MACðφ

~ i φi Þ2 ðφ ~ φ ~ i ÞðφTi φi Þ ðφ T

T i

ð26Þ

~ and φi denote the ith estimated and theoretical modeshape vectors, respectively; unit indicates perfect correlation where φ while zero means no correlation. Step 3: from the knowledge of qi A ℝM (i ¼1,2,3) and R, CS recovers qi A ℝN by the ℓ1 -minimization as per Eqs. (23) to (25). Frequency and damping ratio are obtained by conducting FT and logarithm decrement on the recovered qi A ℝN , respectively. 6.2. Performance results 6.2.1. Free vibration Free vibration is induced by initial conditions of the structures with different damping levels. It is seen from Table 2 that the MAC values are very close to perfect correlation, indicating that the estimated modeshapes are very accurate (Fig. 4(a)). From the M CP-decoupled non-uniform low-rate random samples of each estimated modal response, the uniform high-rate modal response is recovered as per Eqs. (23)–(25). Fig. 5(a) shows the solution found by ℓ1 -minimization of the frequency representation of the modal response is indeed very sparse. Fig. 5(b) shows the recovery results for β ¼0.01 (Fig. 6 is for β ¼0.05). It can be seen that the recovered modal responses are the desired exponentially decaying sinusoids that are monotone with clear envelops, from which frequency and damping ratio are estimated. Table 2 summarizes the identification results, which are accurate when compared with the theoretical ones, regardless of the damping levels. Discussion: Note that the average under-sampled factor N/M is about 6, this is because in practice the modal response can never be ideally K ¼1 sparse, due to numerical errors (DCT, energy leakage, or noise, etc). However, M⪡N holds and the M measurements are randomly selected, where the instantaneous sampling rate can be much smaller than what is required by

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Fig. 3. Performing Step 1 non-uniform low-rate random sensing: (a) the numerical model of 3-DOF linear mass-spring damped structure; and (b) the M ¼ 400 non-uniform low-rate random samples measured by CS of the N ¼ 2000-dimensional displacements (β ¼ 0.01). (A close-up version of the 10–20 s is shown in Step 1 of Fig. 2.).

Table 2 Identification results by CP–CS in free vibration. β

N

M

0.01

2000

400

0.05

1000

180

0.10

800

120

Mode

1 2 3 1 2 3 1 2 3

Frequency (Hz)

Damping ratio (%)

Exact

Recovered

Exact

Recovered

0.0895 0.1458 0.2522 0.0895 0.1458 0.2522 0.0895 0.1458 0.2522

0.0879 0.1465 0.2539 0.0879 0.1465 0.2539 0.0879 0.1465 0.2539

0.8887 0.5460 0.3116 4.4437 2.7299 1.5775 8.8874 5.4598 3.1550

0.8538 0.5645 0.3064 4.3543 2.6953 1.3914 8.6389 5.2288 3.0709

MAC

0.9967 0.9979 0.9998 0.9922 0.9870 0.9999 0.9732 0.9246 0.9981

the Nyquist sampling theorem. In this non-uniform low-rate random sensing example, the largest instantaneous sampling interval is as large as 40, which results in a lowest instantaneous sampling rate of 10 Hz/40 E0.25 Hz (X A ℝ3N at a uniform rate of 10 Hz). This has been below the Nyquist rate of uniform sampling, because capturing the 3rd mode of  0.25 Hz would require at least  0.5 Hz uniform sampling rate to avoid aliasing according to the Nyquist sampling theorem.

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Fig. 4. The results performing Step 2 CP blind decoupling: (a) The estimated modeshapes of the 3-DOF numerical model by CP directly from the M ¼400 non-uniform low-rate random samples of structural responses (N¼ 2000-dimensional); and (b) the CP-decoupled M ¼400 low-rate random samples of the estimated modal response (β ¼0.01). (A close-up version of the 10–20 s is shown in Step 2 of Fig. 2.).

6.2.2. CP–CS compared with ICA & SOBI in the CS framework The assumption of source independence enables BSS to handle the non-uniform low-rate random sensing situation. It has been shown in Section 4.2 that the CP learning rule holds in the non-uniform low-rate random sensing situation for modal identification. For comparisons, the effectiveness of another two BSS methods, namely, ICA and SOBI, is also examined here. ICA uses non-Gaussianity to measure source independence [8]. It is seen in Table 3 that ICA works well in both Nyquist uniform sensing (M¼N) and non-uniform low-rate random sensing (M⪡N) when the damping is low (β ¼0.01). However, higher damping (β ¼0.05) causes it to fail in both uniform sensing and low-rate random sensing. This is consistent with previous studies in Nyquist uniform sensing [14,18], which have shown that ICA is limited to identification of very lightlydamped structures. SOBI uses the second-order statistics of the signals and jointly diagonalizes the covariance matrices with a few time-lags [10]. In the examples, 20 time-lags are used in its joint-diagonalization implementation algorithm. The results in Table 3 show that SOBI works very well in the Nyquist uniform sensing cases. However, in random sensing cases, it shows some degradation. For example, when β ¼0.05, the accuracy of the estimation of the 2nd mode degrades when M decreases from 600 to 180. Similar results occur when β ¼0.10.

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Fig. 5. The results performing Step 3 sparse recovery: (a) from the knowledge of the sensing matrix R A ℝNM and the CP-decoupled M ¼400 non-uniform low-rate random samples of the estimated modal response (shown in Fig. 4(b)), ℓ1 -minimization recovers (using Eq. (23)) the sparse solution αni A ℝN (N ¼2000, each solution corresponds to one mode) in the frequency domain; and (b) the recovered (using Eq. (25)) uniform high-rate modal response, where frequency and damping ratio can be estimated by Fourier transform and logarithm decrement, respectively (results are shown in Table 1) (β¼ 0.01). (A close-up version of the 10–20 s is shown in Step 3 of Fig. 2.).

It is also seen from Table 3 that CP works well in most cases whether in uniform sensing or low-rate random sensing. This indicates that due to the independence assumption of the BSS techniques, the BSS techniques generally have similar performance in non-uniform low-rate random sensing situation as compared to those in Nyquist uniform sensing. 6.2.3. Noise effects and comparison with Nyquist uniform CP identification The performance of the proposed method under the influence of noise is investigated in this section. The case of β ¼0.01 is considered for example, and zero-mean Gaussian white noise is added to the system responses. The noise level is measured by signal-to-noise ratio (SNR), SNR ¼ 20log10

RMSðsignalÞ RMSðnoiseÞ

ð27Þ

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Fig. 6. From M¼ 180 non-uniform low-rate random samples, in Step 3 CP–CS recovers the N ¼ 1000-dimensional uniform high-rate modal response (left column) (β ¼ 0.05); the right column is their power spectral density (PSD).

Table 3 MAC results in free vibration of ICA, SOBI, and CP. ICA

SOBI

CP

Mode

N

M

1

2

3

1

2

3

1

2

3

β ¼ 0:01

2000 2000 1000 1000 1000 800 800 800

2000 400 1000 600 180 800 250 120

1.0000 0.9953 0.7192 – – – – –

0.9999 0.9969 0.2990 – – – – –

1.0000 0.9889 0.4820 – – – – –

0.9999 0.9968 0.9992 0.9974 0.9867 0.9965 0.9882 0.9771

0.9999 0.9641 0.9989 0.9840 0.8629 0.9958 0.9068 0.8029

1.0000 0.9904 0.9997 0.9967 0.9525 0.9982 0.9776 0.8908

1.0000 0.9967 0.9990 1.0000 0.9922 0.9958 0.9944 0.9732

0.9997 0.9979 0.9993 0.9869 0.9870 0.9925 0.9833 0.9246

1.0000 0.9998 0.9974 0.9973 0.9999 0.9918 0.9976 0.9981

β ¼ 0:05 β ¼ 0:10

Table 4 MAC in noisy free-vibration and compared to uniform CP (N ¼2000, β ¼0.01). SNR

20 dB

10 dB

Mode

1

2

3

1

2

3

M¼ 400 M¼ N

0.9933 0.9956

0.9502 0.9425

0.9992 0.9980

0.9854 0.8943

0.9485 0.9708

0.9966 0.8969

where RMS denotes the root-mean-square of a signal. It is seen in the Table 4 that when SNR¼20 dB (10% RMS noise), CP performs well on the M¼400 non-uniform low-rate random measurements, with high-fidelity modeshapes, and the recovered modal responses are reasonable (Fig. 7) and similar to those from the uniform sensing (M¼ N). When the noise increases to as heavy as SNR¼10 dB (31.6% RMS level), the CP performance (MAC results) in non-uniform low-rate random sensing is better than that in Nyquist sensing. It seems that the proposed method works well in the presence of reasonable noise level.

6.2.4. Comparison with direct CS recovery (CS–CP) As mentioned at the end of Section 4.1, after sampling M non-uniform low-rate random measurements of the system response, one may use ℓ1 -minimization to directly recover the N-dimensional system responses, upon which CP can then

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Fig. 7. From M ¼400 non-uniform low-rate random samples, in Step 3 CP–CS recovers the N¼ 2000-dimensional uniform high-rate modal response (left column) (β¼ 0.01) in noisy environment (SNR ¼20 dB); the right column is their power spectral density (PSD).

Table 5 MAC results by direct CS recovery (CS–CP), β ¼ 0:01. SNR

N

No noise No noise 20 dB 10 dB 10 dB

2000

M

Mode 1

Mode 2

Mode 3

650 650 650 1500

0.9896 0.9985 0.9941 0.8496 0.8574

0.6805 0.9966 0.8430 0.9670 0.9639

0.9953 0.9992 0.9970 0.8081 0.8574

Table 6 MAC results by CP–CS in random vibration.

β

N

M

0.01 0.05 0.10

2000

400

Excitation

Mode 1

Mode 2

Mode 3

Stat. Non-stat. Stat. Non-stat. Stat. Non-stat.

0.9990 0.9778 0.9985 0.9778 0.9984 0.9527

0.9858 0.9763 0.9971 0.9763 0.9851 0.9905

0.9953 0.9983 0.9991 0.9983 0.9988 0.9999

perform modal identification—this is termed CS–CP, as opposed to the proposed CP–CS. Such an alternative is conducted for the case β ¼0.01 with N ¼2000, and the MAC results with different situations are presented in Table 5. It is seen that when there is no noise, using M ¼400 is insufficient for CS recovery until M is increased to 650. The situation is worse when polluted by noise. For SNR ¼10 dB, this alternative strategy shows degradation, which may not be made up for by increasing M even to 1500. In comparison, the proposed CP–CS method performs well (Tables 2 and 4), indicating that the proposed CP–CS method is superior to CS–CP. 6.2.5. Random vibration This section studies the effectiveness of the proposed method in random vibration. Stationary random excitation is modeled by zero-mean and unit-variance Gaussian white noise, while non-stationary excitation modulates the stationary one by an exponential function with a constant decaying rate. Because the random vibration will not decay as in free vibration, for different damping levels, the structural response has N ¼ 2000 dimension and CS randomly samples M¼400 non-uniform low-rate measurements. Table 6 shows the results, indicating that the CP–CS method is also effective in

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random vibration. This is because the CP learning rule is valid even in random vibration, and the modal responses in random vibration hold sparsity in the frequency domain for CS to accurately recover (Fig. 8). 7. Simulation study on an experimental structure model and a real-world seismically-excited building Although the prototype of the non-uniform low-rate CS based sampler has been produced very lately [27–30], (as well as vast novel CS hardware instruments [34,35]), currently, there are no CS sensors commercially available; therefore, the

Fig. 8. From M¼ 400 non-uniform low-rate random samples, in Step 3 CP–CS recovers N ¼ 2000-dimensional uniform high-rate modal response (left column) in random vibration (β ¼ 0.01); the right column is their power spectral density (PSD).

Fig. 9. The experimental bench-scale 3-story steel frame.

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functionality of CS sensors is simulated (in MATLAB) in the study of the proposed CP–CS method conducted on a three-story steel frame (Fig. 9) experimental bench-scale structural model and a real-world seismically-excited building (Fig. 10) to demonstrate its applicability. In Step 1 data acquisition stage, at each channel, the CS sensor randomly senses M⪡N non-uniform low-rate samples. The data analysis steps are then performed on the available non-uniform low-rate random samples. It is important to note that if a non-uniform CS sampler is available in practice, it directly senses the M⪡N random measurements; see the non-uniform CS sampler paradigm proposed in Ref. [7] and its prototype in Ref. [27].

Fig. 10. (a) The estimated modeshapes of the experimental 3-story structure by CP–CS in non-uniform low-rate random sensing framework, comparing to those by CP and peak picking (PP) in Nyquist uniform sensing framework; and (b) from M ¼1023 non-uniform low-rate random samples (M EN/10), CP–CS recovers the N ¼ 10,226-dimensional uniform high-rate modal response; the right column is their power spectral density (PSD).

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7.1. Experimental bench-scale structural model The experimental bench-scale structural model has three dominant masses on the floors and two steel columns. The base of this model is fixed on a small shaking table with a feedback control system. White noise excitation is generated by the shaking table, exciting the structure. The structure is outputting N ¼10,226-dimensional structural response accelerations at each of the three channels, then in Step 1 data acquisition the CS sensor randomly senses only M ¼1023 (N/M E10) non-uniform low-rate samples X A ℝ31023 . For the data analysis stage, CP is then directly performed on X A ℝ31023 , simultaneously yielding the mode shapes and the non-uniform low-rate random samples of the estimated modal responses Q A ℝ31023 , and then the ℓ1 -minimization is used to recover the uniform high-rate modal response qi A ℝ10;226 (i¼1,2,3), whereby estimating modal information (frequency). Since this study focuses on the effectiveness of the output-only CP–CS, estimation of the damping ratio is not conducted herein because it would require additional preprocessing (e.g., the random decrement technique) for reliable estimation. The identification results are summarized in Table 7 and Fig. 10(b) shows the recovered uniform high-rate modal responses by the proposed CP–CS. For comparisons, the CP in Nyquist uniform sampling (at 160 Hz) is conducted on X A ℝ310;226 . Besides, the classical input-output peak-picking (PP) method (in Nyquist uniform sampling framework) was previously conducted on this structural model by the authors in Ref. [21] and the results are also used for comparison reference. The computations of the MAC values both use the modeshapes estimated by PP method as reference. It is seen that the frequency estimated by these three methods matches well, and the high MAC values indicate the correlations of the modeshapes between the PP and CP–CS and between PP and CP are quite good (Fig. 10(a)). Discussions: With comparable accuracy with traditional methods, the advantage of the proposed CP–CS method is evident. In this example, the non-uniform random sensing rate is on average 160 Hz/10 ¼16 Hz (because N/M E10) (the lowest instantaneous sampling rate is 2.50 Hz). On the other hand, the Nyquist sampling theorem indicates that capturing the 3rd mode of more than 10 Hz would require a uniform sensing rate at least 20 Hz to avoid aliasing (in practice, this would be much higher). Using the non-uniform low-rate random sensing framework, the (average) sampling rate can be below what is required by the sampling theorem, sensing much less samples in the data acquisition stage.

7.2. A seismic-excited structure example Another example is also presented to demonstrate the applicability of the proposed CP–CS method. The University of Southern California (USC) hospital building is an eight-story base-isolated nonlinear structure with high damping (the 1st mode damping ratio is about 14%) [41], and experienced the 1994 Northridge earthquake. Its sensor outline is shown in Fig. 11. At each of the three channels (the #12, 17, and 21 sensors in the north–south direction), the building is outputting N ¼1500-dimensional structural seismic response accelerations (Nis small due to the short duration of earthquake excitation), then in Step 1 data acquisition the CS sensor randomly senses only M¼100(N/M ¼15) non-uniform low-rate samples X A ℝ3100 (which are directly measured using a non-uniform CS sampler [27–30] in practice when commercially available). Step 2 and Step 3 of the proposed CP–CS method are then applied on X A ℝ3100 . A 3D analytical model of this structure was previously developed in Ref. [41] and the analytical results of that study are used for comparison reference. Also, the authors have applied the CP on the uniform (at 100 Hz) X A ℝ31500 in Ref. [21] and the results are presented herein to compare with those from the non-uniform low-rate random sensing. It is seen in Table 8 and Fig. 12(a) that the MAC (using the modeshapes of the analytical model as reference) of the low-rate random sensing case shows some degradation compared to that by the high-rate CP case; however, the estimated frequencies (Table 8 and Fig. 12(b)) are the same with those by CP in the Nyquist sampling case and match the analytical FEM results. Discussion: It is seen that the average sampling rate of the non-uniform low-rate random sensing method has been 100 HHz/15 ¼6.67 HHz (because N/ME15) (the lowest instantaneous sampling rate is 1.75 Hz), which is below the required sampling frequency of the 3rd mode (the sampling theorem would require at least  7.4 Hz, much higher in practice, to avoid aliasing). This indicates that using the proposed method, it is feasible to sense much fewer samples at a sub-Nyquist (average) rate to obtain reasonably accurate modal identification results. Also the lower estimated MAC accuracy of the 3rd modeshape is primarily because this mode is rarely excited, which has already been discussed in Ref. [18,21]. Table 7 MAC results by CP–CS of the experimental model (M  N=10). Mode

1 2 3

Frequency (Hz)

MAC

PP

Uniform CP

CP–CS

Uniform CP

CP–CS

2.550 7.330 10.460

2.600 7.395 10.720

2.602 7.395 10.720

1.0000 0.9996 0.9983

0.9963 0.9989 0.9669

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Fig. 11. The USC hospital building.

Table 8 Identification results by CP–CS of the USC building (M ¼ N=15). Mode

1 2 3

Frequency (Hz)

MAC

FEM

Uniform CP

CP–CS

Uniform CP

CP–CS

0.746 1.786 3.704

0.768 1.907 3.941

0.768 1.907 3.941

0.9751 0.9054 0.7874

0.9249 0.7013 0.5311

8. Conclusions and future study This study presents a new output-only modal identification method based on a combination of CS and BSS (CP) using the non-uniform low-rate random sensing framework. It provides an alternative to the traditional Nyquist uniform sensing by alleviating the potentially high sampling rate and requiring much less sample rate (hence much less samples need to be acquired in the data acquisition step) for modal identification. It is found that the modal identification problem naturally satisfies the requirements of CS: the modal response has sparsest representation in the frequency domain, which is maximally incoherent with the spike sensing space. Also, the independence assumption of BSS is exploited and found to be suitable in the non-uniform low-rate random sensing situation. Study on a numerical structural model, an experimental bench-scale structural model, and a real-world seismicexcited structure, where the functionality of the CS sensor is simulated, are presented for demonstrating the effectiveness of the proposed CP–CS method. Results show that the proposed CP–CS method can capture the modes with much fewer lowrate random samples or lower sampling rate than what is required by the traditional Nyquist sampling theorem. It is also shown to be robust to noise. Although this study focus on investigation of its feasibility for output-only modal identification, it may also be applied to traditional experimental modal analysis (FRF measurements) using the non-uniform low-rate random sensing and ℓ1 -minimization techniques—which is the subject of ongoing study by the authors. In addition, recent study [27–30] has shown that the CS samplers (in laboratory prototype) are more resource-efficient (e.g., power) than traditional Nyquist samplers; when they become commercially available, it is also of practical interest to examine their efficiency in applications of modal identification and analysis of structures (e.g., in wireless platform) in future study.

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Fig. 12. (a) The estimated modeshapes of the USC building by CP–CS in non-uniform low-rate random sensing framework, comparing to those by CP in Nyquist uniform sensing framework and to those by FEM; (b) from M ¼100 non-uniform low-rate random samples (M¼ N/15), CP–CS recovers N ¼1500dimensional uniform modal responses.

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