Underdetermined modal parameter identification for time-invariant structures via autocorrelation-optimized CP decomposition

Abstract

In recent years, operational modal parameter identification has become crucial in fields such as engineering, aerospace, mechanics, and civil infrastructure, as well as in analyzing complex structures. However, when the number of sensors is fewer than the modal parameters to be identified, the system becomes underdetermined, meaning that the limited sensor data cannot fully capture the system’s modal information. This often results in uncertainties and challenges in the identification process. Current methods for underdetermined operational modal parameter identification typically rely on signal sparsity or statistical independence assumptions, often requiring signal preprocessing to enforce sparsity, which restricts their applicability. To address these limitations, this study presents a novel underdetermined operational modal parameter identification method based on autocorrelation-optimized principal component tensor decomposition. This approach first establishes a connection between operational modal parameter identification and CP tensor decomposition, then introduces a new tensor construction technique that combines statistical characteristics of the data with time-series analysis. By decomposing the constructed tensor, the method extracts the modal shape matrix along with a set of sub-tensors representing modal responses, enabling the estimation of modal parameters. The method’s effectiveness is verified through experiments on a three-degree-of-freedom spring oscillator and a uniform cantilever beam structure, even under underdetermined and noisy conditions. Results demonstrate that the proposed method outperforms existing techniques in both recognition accuracy and noise resistance, providing a stable and precise identification of operational modal parameters.

Keywords

Structural dynamics underdetermined operational modal parameter identification tensor decomposition autocorrelation optimization tensor construction

1. Introduction

In industries such as aerospace, automotive, maritime, railways, and heavy machinery, vibration sensitivity is a critical concern. With the increasing scale and dynamic loads on engineering structures (Taira et al., 2020), accurate modal parameter identification has become essential for ensuring structural safety and maintainability. Modal analysis is commonly used for quality control and structural health monitoring, (Rehman et al., 2024; Giordano et al., 2023; Nicoletti et al., 2023) providing vital data for managing potential failures.

Operational Modal Analysis (OMA) (Qin et al., 2024; Civera et al., 2023) derives modal parameters from response signals and has become widely used in structural monitoring across fields such as wind turbines, high-speed aircraft, and automotive engines (Zahid et al., 2020). However, as engineering structures become faster (Li et al., 2023), larger, and more complex, obtaining complete and accurate modal information is challenging, especially when sensor placement is limited (Feng and Feng, 2018). This limitation creates underdetermined conditions (Guan et al., 2019), where the data from a few sensors fail to capture the system’s full dynamic response, hindering effective modal analysis (Cadoret, 2023).

To address underdetermined identification, two primary approaches have emerged: sparsity-based algorithms (Yu et al., 2014) and tensor decomposition methods (Guan et al., 2021). Sparse Component Analysis (SCA) (Yao et al., 2022; Liu et al., 2020), which leverages signal sparsity in the frequency or time-frequency domain (Karami et al., 2020), has shown promise. By applying various transformations (e.g., Short-Time Fourier (Zhang and Deng, 2023), Discrete Cosine (Yang and Nagarajaiah, 2013), Wavelet Packet Transforms (Li et al., 2016)), SCA allows sparse signal representation, followed by clustering techniques (Zhou et al., 2018) to estimate mode shapes. However, the accuracy of this method heavily depends on clustering quality (Yao et al., 2018), which may lead to loss of modal information if suboptimal (Sadhu et al., 2013).

Tensor decomposition methods, such as those based on the Parallel Factor Analysis (PARAFAC) framework (Friesen and Sadhu, 2017), offer high computational efficiency and avoid the need for independence or sparsity assumptions. Recent advances using Block Term Decomposition (BTD) (Guan et al., 2021) have enabled accurate mode identification in limited-sensor settings by leveraging low-rank signal representations (Greś et al., 2021). Nevertheless, these methods face several challenges (Bousse et al., 2016). First, there is a need to optimize decomposition parameters because the choice of rank and other hyperparameters in Block Term Decomposition (BTD) directly impacts the accuracy and generalization of the model. Without proper optimization, the decomposition may fail to capture the essential modes of the data or lead to overfitting (Le et al., 2024; Li et al., 2016). This optimization process can be computationally expensive and requires careful tuning to adapt to different datasets or applications. Second, the limitations of static tensor construction pose another significant challenge. In many real-world scenarios, the data is not stationary and exhibits temporal or dynamic characteristics, such as in time-series or sequential data. Static tensor methods assume a fixed structure of the data, which may overlook important time-dependent features and non-stationary data characteristics. For example, in industrial systems, the behavior of the system can change over time due to external influences or system degradation, which static methods cannot adapt to. As a result, these methods may fail to accurately represent or predict the system’s behavior over time, leading to suboptimal mode identification or poor predictive performance (Koutsoukos et al., 2021).

Tucker decomposition (Shao et al., 2022) is a widely used tensor factorization method that decomposes high-dimensional tensors into the product of multiple matrices and a core tensor, suitable for low-rank approximation of high-order tensors (Xie et al., 2023). However, it typically does not account for temporal variations and dynamics in the data, and performs poorly in scenarios with high noise or limited data. PARAFAC2 is an extension of PARAFAC, designed to handle time-series data by considering non-stationarity between mode dimensions. However, it still faces challenges in dynamic mode identification, particularly when there are multiple modes or high noise, and especially when there is strong correlation between modes. BTD (Block Term Decomposition) is usually based on covariance matrix decomposition, but it does not fully leverage the temporal autocorrelation information within tensor data. As a result, it struggles to accurately identify dynamic modes, particularly when there are complex temporal dependencies in the data.

Although existing covariance-based methods (Pan et al., 2021) can handle underdetermined problems, they rely on the decomposition of covariance matrices, and the multiple decomposition steps may lead to reduced accuracy, particularly in the presence of high noise or weak mode signals. Additionally, covariance methods are heavily reliant on global feature modeling of the system, making it difficult to effectively capture subtle temporal correlations between modes, resulting in poor performance in complex underdetermined problems. While Bayesian CP decomposition (Giampouras et al., 2022) can automatically select the number of modes and handle underdetermined systems, its effectiveness is limited in cases where modes are strongly correlated or sparse. In particular, when mode signals are weak or data is scarce, Bayesian inference may struggle to accurately identify all active modes. In contrast, the method proposed in this paper enhances mode identification by optimizing autocorrelation, particularly in underdetermined cases where it can effectively reduce noise interference and improve the accuracy of mode parameter identification. Autocorrelation optimization is better equipped to address the limited data problem, improving the accuracy of low-rank tensor decomposition, and ensuring that useful mode features can still be extracted even with insufficient data. Therefore, compared to Bayesian CP decomposition and covariance methods, the proposed method demonstrates stronger robustness and higher identification accuracy in the domain of mode parameter identification for underdetermined systems. This study aims to overcome these limitations by introducing a dynamic tensor construction approach that adapts to data changes and enhances robustness in real-world applications, where non-stationary characteristics are common.

A linear time-invariant underdetermined operating mode parameter identification method based on autocorrelation optimized principal component CP decomposition (APCA-CPD) is proposed. By integrating autocorrelation, Principal Component Analysis (PCA) (Maćkiewicz and Ratajczak, 1993), and CP decomposition (Kolda and Bader, 2009), the method aims to explore and analyze complex structures and dynamic characteristics within multidimensional time series data, thereby enabling better identification of modal parameters. The contributions of this paper are summarized as follows:

(1) By calculating the autocorrelation function and identifying its peaks, this method captures the periodic characteristics of the data corresponding to the structure’s natural frequencies. The lag number is then determined from the intervals between these peaks, enabling a more accurate representation of the data’s dynamic structure.

(2) To overcome the limitations of static tensor construction, this study proposes a method to dynamically adjust the lag number and tensor dimensions based on the data’s statistical properties. This approach optimizes tensor construction to capture key dynamic features more effectively.

(3) In scenarios with limited sensors, traditional multidimensional analysis may not fully utilize the data’s richness. By incorporating the lagged perspective of time-series data, APCA-CPD enhances the data’s information capacity, enabling effective extraction of modal information even with fewer sensors.

The remainder of this paper is organized as follows: “Related Work” provides an overview of operational modal parameter identification for linear time-invariant structures and its connection to CP decomposition. “Proposed Method” details the methodology. “Experimental Results and Discussion” presents datasets, evaluation methods, and results, including comparisons, ablation studies, and noise robustness tests. Finally, the “Conclusion” summarizes the study and discusses future research directions.

2. Related work

2.1. The basic theory of OMA parameter identification for LTI Structures

The core of modal parameter identification lies in transforming vibration differential equations from the physical to the modal coordinate system, thereby decoupling them into multiple independent single-degree-of-freedom (SDOF) systems. In the modal coordinate system, modal vectors are orthogonal and uncoupled, allowing for the determination of key modal parameters—such as mode shapes, frequencies, and damping ratios—using SDOF identification techniques (Fan and Qiao, 2011). For complex structural vibrations, engineering analysis typically simplifies the structure from an infinite to a finite degree-of-freedom system. This study, therefore, uses a multi-degree-of-freedom (MDOF) vibration system as a representative example to address the modal parameter identification problem.

The vibration differential equation of an n-degree-of-freedom (DOF) linear time-invariant system is expressed as follows:

M \ddot{X} (t) + C \dot{X} (t) + K X (t) = F (t)

(1)

Among them,

M

、

C

、

K

are the time-invariant mass, damping, and stiffness matrices, respectively.

X (t)

is the displacement response vector of the multi-degree-of-freedom system,

\dot{X} (t)

is the velocity response vector,

\ddot{X} (t)

is the acceleration response vector,

F (t)

is the external excitation vector.

According to the theory of vibration dynamics, the natural frequencies and mode shapes of an n-degree-of-freedom linear system can be obtained using the following equations:

| K - w_{i}^{2} M | {\vec{φ}}_{i} = 0

(2)

The arithmetic square root of the eigenvalue $ω_{i}^{2}$ is the natural frequency $ω_{i}$ of the system, and the non-zero eigenvector ${\vec{ϕ}}_{i}$ corresponding to $ω_{i}$ that satisfies equation (2) is the system’s $i$ -th mode shape.

To aid in understanding this paper, the notation system used is provided in Table 1 for quick reference.

Table 1.

Global symbol table.

Symbol	Meaning description	Specification
$n_{m}$	Number of sensors
N	The number of data points in the sensor
$R$	Real number $n_{d}$ Degree of freedom
$M$	Mass matrix of vibration system	$R^{n_{m} \times n_{m}}$
$C$	Damping matrix of vibration system	$R^{n_{m} \times n_{m}}$
$K$	Stiffness matrix of vibration system	$R^{n_{m} \times n_{m}}$
$X (t)$	Displacement response signal	$R^{n_{m} \times N}$
$\dot{X} (t)$	Velocity response signal	$R^{n_{m} \times N}$
$\ddot{X} (t)$	Acceleration response signal	$R^{n_{m} \times N}$
$ω_{i}$	The i-th natural frequency
${\vec{φ}}_{i}$	The i-th true mode shape of the system	$R^{n_{m} \times 1}$
$ξ_{i}$	Modal damping ratio
$Φ$	Mode mode matrix	$R^{n_{d} \times n_{m}}$
$Λ$	Diagonal matrix of eigenroots	$R^{n_{m} \times n_{m}}$
$F (t)$	Excitation of external load	$R^{n_{m} \times N}$
$Q (t)$	Modal response matrix	$R^{n_{d} \times N}$
$τ_{k}$	Lag
$Z_{X}$	Covariance matrix of vibration measurements $X$	$R^{n_{m} \times n_{m}}$
$Γ$	Modal-to-measurement transformation matrix $R^{n_{m} \times n_{d}}$
$°$	Outer product of vectors
$\vec{x}$	Tensor of first order	$R^{n_{1}}$
$X$	Second order tensor	$R^{n_{1} \times n_{2}}$
$\underline{X}$	Tensor of third order	$R^{n_{1} \times n_{2} \times n_{3}}$
R	Number of rank-1 tensors
e	Khatri–Rao product
$†$	Moore–Penrose generalized inverse
g	Cost function
$\underline{\hat{X}}$	The optimal estimation tensor	$R^{n_{m} \times N \times τ_{k}}$
B	The matrix after the original data is normalized	$R^{n_{m} \times n_{m}}$
$μ$	The mean of the original data in each measurement channel	$R^{n_{m} \times 1}$
$σ$	Standard deviation of each channel	$R^{n_{m} \times 1}$
Y	Principal component score matrix	$R^{n_{m} \times n_{m}}$
R(l)	Autocorrelation function locs Peak array
f	Sampling frequency
${\vec{ϕ}}_{i}$	The estimated i-th mode shape vector	$R^{n_{m} \times 1}$
${MAC}_{i}$	The i-th modal confidence value
$ω_{i - t h e o r y}$	The i-th theoretical natural frequency
$ζ_{i}$	Natural frequency relative error

2.2. The correspondence between OMA and CP decomposition

Consider a linear, classically damped, lumped-mass structural system with $n_{d}$ degrees of freedom (DOF), subjected to an excitation force $F (t)$ .The solution to equation (1) can be represented as a modal superposition of the vibration modes, and its matrix form is expressed as follows (Liu et al., 2020):

X (t) = Φ Q (t)

(3)

where

X (t) \in R^{n_{m} \times N}

is the matrix composed of the sampled data of

X (t)

Q (t) \in R^{n_{d} \times N}

is the matrix of impulse responses corresponding to the modal coordinates, and

Φ \in R^{n_{m} \times n_{d}}

is the mode shape matrix, where n_m is the number of measurement channels (i.e., the number of sensors), N is the number of data points per channel, and

n_{d}

is the number of degrees of freedom.

The covariance matrix $Z_{X} (τ_{k})$ of the vibration measurements $X (t)$ evaluated at time delay $τ_{k}$ can be expressed as:

Z_{X} (τ_{k}) = E {X (n) X^{T} (n - τ_{k})} = Γ Z_{Q} (τ_{k}) Γ^{T}

(4)

where

Z_{Q} (τ_{k}) = E {Q (n) Q^{T} (n - τ_{k})}

(5)

Here, $Z_{X} (τ_{k}) \in R^{n_{m} \times n_{m}}$ and $Z_{Q} (τ_{k}) \in R^{n_{d} \times n_{d}}$ , $Γ \in R^{n_{m} \times n_{d}}$ is the transformation matrix from modal coordinates to measurement coordinates. To simplify the notation, the following definition is used:

Z_{X_{1} X_{2}} (τ_{k}) = Z_{12 k}^{X} \Leftrightarrow Z_{X_{i} X_{j}} (τ_{k}) = Z_{i j k}^{X}

(6)

Z_{Q 1} (τ_{k}) = Z_{k 1}^{Q} \Leftrightarrow Z_{Q l} (τ_{k}) = Z_{k l}^{Q}

(7)

The elements $Z_{i j k}^{X}$ of the covariance matrix $Z_{X} (τ_{k})$ represent the covariance between measurement channels $X_{i}$ and $X_{j}$ at the time delay $τ_{k}$ .this can be generalized into a general form, as shown in equation (8):

\begin{array}{l} Z_{i j k}^{X} = Γ_{i 1} Γ_{j 1} Z_{k 1}^{Q} + Γ_{i 2} Γ_{j 2} Z_{k 2}^{Q} + Γ_{i 3} Γ_{j 3} Z_{k 3}^{Q} \\ = \sum_{r = 1}^{3} Γ_{i r} Γ_{j r} Z_{k r}^{Q} i, j = 1 : 3; k = 1 : τ_{k} \end{array}

(8)

for a general

n_{d}

-DOF dynamic system, equation (8) can be simplified as:

Z_{i j k}^{X} = \sum_{r = 1}^{n_{d}} Γ_{i r} Γ_{j r} Z_{k r}^{Q} \Leftrightarrow {\underline{Z}}^{X} = \sum_{r = 1}^{n_{d}} Γ_{r} ° Γ_{r} ° Z_{r}^{Q}

(9)

where “

°

”denotes the vector outer product.

A third-order tensor is primarily decomposed into the sum of the outer products of column vectors from three factor matrices, as shown in Figure 1(b):

\underline{X} = \sum_{r = 1}^{R} ϕ_{r} ° q_{r} ° θ_{r} \Leftrightarrow {\underline{X}}_{i j k} = \sum_{r = 1}^{R} ϕ_{i r} q_{j r} θ_{k r}

(10)

where

i \in [1, I]

j \in [1, J]

,and

k \in [1, K]

, the elements

ϕ_{i r}

q_{j r}

,and

θ_{k r}

correspond to the elements in the respective vectors

ϕ_{i}

q_{j}

, and

θ_{k}

associated with the tensor’s indices i, j, and k while R is the rank of tensor

\underline{X}

Figure 1.

(a) 2 × 2 × 2 Tensor block representation. (b) Sum of rank-1 tensors.

As illustrated in Figure 1(a), in the case where I = J = K = 2, a single tensor $\underline{X}$ and its elements can be expressed as:

\begin{array}{l} ϕ ° q ° θ = (\begin{array}{l} ϕ_{1} \\ ϕ_{2} \end{array}) ° (\begin{array}{l} q_{1} \\ q_{2} \end{array}) ° (\begin{array}{l} θ_{1} \\ θ_{2} \end{array}) \\ \equiv [\begin{array}{l} ϕ_{1} q_{1} θ_{1} & ϕ_{1} q_{2} θ_{1} & ϕ_{1} q_{1} θ_{2} & ϕ_{1} q_{2} θ_{2} \\ ϕ_{2} q_{1} θ_{1} & ϕ_{2} q_{2} θ_{1} & ϕ_{2} q_{1} θ_{2} & ϕ_{2} q_{2} θ_{2} \end{array}] \\ \equiv [\begin{array}{l} x_{111} & x_{121} & x_{112} & x_{122} \\ x_{211} & x_{221} & x_{212} & x_{222} \end{array}] \end{array}

(11)

Equation (10) and Figure 1(b) represent the sum of R CP components of a high-order tensor $\underline{X}$ .

By observing equations (9) and (10), it can be seen that the two equations are similar. Therefore, the third-order tensor ${\underline{Z}}^{X}$ can be decomposed into $n_{d}$ CP components. Through CP decomposition of ${\underline{Z}}^{X}$ , the mixing matrix (i.e., mode shape matrix) $Γ = [Γ_{1}, Γ_{2}, Γ_{3} \dots, Γ_{n_{d}}]$ and the associated matrix $Z_{r}^{Q}$ , $r = 1, 2, 3, \dots, n_{d}$ , can be estimated to obtain the natural frequencies and damping ratios corresponding to each mode.

Through the above brief introduction, the mathematical equivalence between modal identification in the context of operational modal analysis and CP decomposition has been established (Sadhu, 2013).

3. Methodology

First, Principal Component Analysis (PCA) is applied to the matrix $X (t) \in R^{n_{m} \times N}$ composed of the sampled data to analyze and extract the most important features from the original data.

First, standardize the data collected from the sensors:

B = (X - μ) / σ

(12)

where $μ \in R^{n_{m} \times 1}$ represents the mean value of the original data for each measurement channel, and $σ \in R^{n_{m} \times 1}$ is a vector containing all ones, ensuring that each measurement channel’s data is centered.

Next, calculate the covariance matrix of the standardized data:

Z = \frac{1}{n - 1} B^{T} B

(13)

where n represents the number of samples, and this matrix is used to perform eigenvalue decomposition on the standardized data.

Then, eigenvalue decomposition is performed on the covariance matrix to obtain:

Z = V Λ V^{T}

(14)

where V is a matrix composed of eigenvectors, and

Λ

is a diagonal matrix containing the corresponding eigenvalues.

Calculate the principal components to obtain the matrix:

Y = B V

(15)

where Y represents the data in the principal component space, with each column being a principal component.

Then, use the significant peaks in the autocorrelation function of the principal components to determine the lag $τ_{k}$ , select a principal component $Y_{(:, i)}$ for autocorrelation function analysis:

R (l) = \frac{\sum_{t = 1}^{N - l} (Y_{t, i} - Y_{i}) (Y_{t + l, i} - Y_{i})}{\sum_{t = 1}^{N} {(Y_{t, i} - Y_{i})}^{2}}

(16)

where N is the length of the data, l is the number of lag points, t is the sampling time, i is the index of the selected principal component (i.e., the i-th column in the principal component matrix),

Y_{t, i}

represents the value of the i-th principal component at time t, and

{\bar{Y}}_{i}

is the mean of

Y_{t, i}

Using the autocorrelation function R(l), the appropriate lag $τ_{k}$ for detecting periodic characteristics is determined by peak detection:

\begin{array}{l} l o c s = {k | A C F (l) > A C F (l - 1) \\ and A C F (l) > A C F (l + 1)} \end{array}

(17)

τ_{k} = {\begin{cases} l o c s [2] - l o c s [1] & i f l e n (l o c s) > 1 \\ l e n (A C F) & o t h e r w i s e \end{cases}

(18)

where locs [2] represents the position of the second peak in the (locs) array, and locs [1] represents the position of the first peak. The len (locs) is the length of the locs array, that is, the number of detected peaks. len (ACF) represents the length of the autocorrelation function, that is, the number of autocorrelation values. Therefore, if at least two peaks are found, the lag

τ_{k}

is the distance between the first and second peaks. If not enough peaks are detected, the maximum length of the autocorrelation function is used as the lag.

Using the determined lag, a time-varying three-dimensional tensor is dynamically constructed, where each “slice” contains the data for a time lag. Let $X (t)$ be the original data matrix, and $τ_{k}$ be the determined lag, a three-dimensional tensor $\underline{X}$ can be constructed with time lag as its third dimension.

The construction of the three-dimensional tensor follows these steps:

1) Initialize the tensor: Start by initializing a three-dimensional tensor $\underline{X}$ , with dimensions $n_{m} \times (N - τ_{k} + 1) \times τ_{k}$ , where $n_{m}$ represents the number of measurement channels (e.g., sensors), and N is the length of the data.

2) Lag embedding: For each time lag from 1 to $τ_{k}$ , extract the corresponding two-dimensional slice as:

\underline{X} [:, :, l] = X [:, l : (N - τ_{k} + l + 1)]

(19)

where

X [:, l : (N - τ_{k} + l + 1)]

represents the data segment extracted from the original data matrix X starting from l with a length of

N - τ_{k} + 1

This tensor construction method allows for flexible adjustment of the lag, dynamically optimizing the tensor’s structure. By utilizing the statistical properties of the data, it captures dynamic, time-dependent features that traditional static tensor constructions might miss. The resulting three-dimensional tensor can capture multichannel data comprehensively while embedding lag along the time sequence dimension (Runge, 2020). This approach is especially effective in extracting modal information from multiple channels, even in challenging scenarios with limited (underdetermined) or noisy data, where traditional methods often fall short.

Finally, the constructed three-dimensional tensor undergoes CP decomposition:

\begin{array}{l} \underline{X} = \sum_{r = 1}^{R} Φ_{r} ° Q_{r} ° Θ_{r} \\ \Leftrightarrow {\underline{X}}_{i j k} = \sum_{r = 1}^{R} Φ_{i r} Q_{j r} Θ_{k r} \end{array}

(20)

where

{\underline{X}}_{i j k}

represents the element at position i, j, and k in tensor

\underline{X}

Φ_{i r}

is the element in the i-th row and r-th column of matrix

Φ

Q_{j r}

is the element in the j-th row and r-th column of matrix Q,

Θ_{k r}

is the element in the k-th row and r-th column of matrix

Θ

; all these components generate the mode shape matrix

Φ = [{\vec{ϕ}}_{1}, {\vec{ϕ}}_{2}, \dots, {\vec{ϕ}}_{R}]

and the modal coordinate matrix

Q = [{\vec{q}}_{1}, {\vec{q}}_{2}, \dots, {\vec{q}}_{R}]

. The corresponding modal natural frequencies

ω_{i}

can be estimated from the modal response matrix. Figure 2 shows the flowchart of the proposed method for identifying operational modal parameters.

Figure 2.

Flow chart of APCA-CPD method for identifying working modal parameters.

4. Experimental verification

4.1. Dataset introduction

In this study, the linear steady three-degree-of-freedom spring vibration subsystem was used as the data set for experimental verification (Kerschen et al., 2007), hereinafter referred to as Dataset 1 (Figure 3). With the help of Simulink/Matlab platform, the Newmark-β integral method is used to solve the dynamic response of the system, and the time-domain response data of vibration is obtained. The time domain displacement response data obtained by simulation were used for simulation verification. More details of simulation can be found in the literature (Wang et al., 2018) With a sampling interval of 0.005s and a frequency of 20 Hz, the simulation runs for 2000s under initial zero conditions. White noise excitation is applied to mass m1, producing the response signal shown in Figure 4. The parameters are set as follows: m1 = m2 = m3 = 1 kg,k = 1000 N/m for stiffness, and C = 0.01 N.s/m for damping. Sensors 2 and 3 capture the system’s response, creating an underdetermined scenario (i.e., $n_{d}$ = 3, $n_{m}$ = 2).

Figure 3.

Model of the linear time-invariant three-degree-of-freedom spring-mass system.

Figure 4.

Gaussian white noise excitation and the obtained response signal.

In this study, the experimental structure of a uniform steel cantilever beam was also adopted as the data set (Yang et al., 2013), hereinafter referred to as Dataset 2, which is an experimental device with a vibrating base and five displacement sensors, the size of which is 0.9 × 0.05 × 0.008 m³. The output signals are sampled digitally by DASP (Data Acquisition & Signal Processing) when the sampling frequency is 1600 Hz, and the cut-off frequency for all the five channels is 800 Hz and the diagram of the experimental device is shown in Figure 5 (Yang et al., 2013). The system is stimulated by an impact hammer with a frame number of 8142 frames. For an underdetermined setup, only sensors 1, 3, and 5 are used (i.e., $n_{d}$ = 5, $n_{m}$ = 3).

Figure 5.

Cantilever beam experimental set.

4.2. Evaluation method

To evaluate the accuracy of estimated modal shapes for a time-invariant structure, this study employs the Modal Assurance Criterion (MAC) as shown in equation (21) (Maia and Montalvão, 1997),

M A C_{i} = \frac{{({\vec{ϕ}}_{i}^{T} {\vec{φ}}_{i})}^{2}}{({\vec{φ}}_{i}^{T} {\vec{φ}}_{i}) ({\vec{ϕ}}_{i}^{T} {\vec{ϕ}}_{i})}

(21)

where

{\vec{φ}}_{i}

and

{\vec{ϕ}}_{i}

represent the true mode shape vector and the estimated mode shape vector for the i-th mode, respectively. The MAC value ranges from 0 to 1, where values closer to 1 indicate greater alignment with the true mode shape and higher identification accuracy, while lower MAC values signify less accurate identification.

Additionally, the identification accuracy of natural frequencies is assessed using the relative error between the identified and theoretical frequencies, calculated as shown in equation (22):

ζ_{i} = | \frac{ω_{i} - ω_{i - t h e o r y}}{ω_{i - t h e o r y}} | \times 100 %

(22)

where

ω_{i}

and

ω_{i - t h e o r y}

represent the identified natural frequency and the theoretical natural frequency of the

i

-th mode, respectively. The closer the value of

ζ_{i}

is to 0, the more accurate the identified natural frequency.

4.3. Contrast experiment

Through both quantitative and qualitative comparisons, the performance of the APCA-CPD method proposed in this study can be comprehensively evaluated on different datasets. Quantitatively, Table 2 presents the comparison of MAC (Modal Assurance Criterion) values between the proposed APCA-CPD method and other benchmark algorithms (such as Tucker decomposition, PARAFAC2, APCA-BTD, and Bayesian CP decomposition) for underdetermined modal parameter identification on Dataset 1. The comparison of MAC values provides an intuitive reflection of the accuracy differences among the methods, with higher MAC values generally indicating a more accurate identification of the operating modes. Furthermore, Table 3 shows the relative errors in the inherent frequencies for Dataset 1, further verifying the superior performance of the proposed method in frequency estimation, as lower relative errors indicate higher frequency estimation accuracy.

Table 2.

Comparison of MAC values for underdetermined modal parameter identification on Dataset 1.

Modal\Algorithm	Tucker	PARAFAC2	Bayes-CPD	APCA-BTD	APCA-CPD
1st	0.9400	0.9963	0.9899	0.9966	1.0000
2nd	0.9482	0.9896	0.9618	0.9761	1.0000
3rd	0.9375	0.8357	0.9153	0.9619	1.0000

Table 3.

Relative error in inherent frequencies for Dataset 1.

Algorithm\Modal		1st	2nd	3rd
Theoretical value (Hz)		2.2399	6.2760	9.0690
Tucker	Identification value	2.2410	6.2720	9.0780
Tucker	Relative error	0.0491%	0.0637%	0.0992%
PARAFAC2	Identification value	2.2420	6.2770	9.1200
PARAFAC2	Relative error	0.0937%	0.0159%	0.5623%
Bayes-CPD	Identification value	2.2430	6.2775	9.0920
Bayes-CPD	Relative error	0.1384%	0.0239%	0.2536%
APCA-BTD	Identification value	2.2420	6.2770	9.0770
APCA-BTD	Relative error	0.0937%	0.0159%	0.0882%
APCA-CPD	Identification value	2.2410	6.2770	9.0800
APCA-CPD	Relative error	0.0491%	0.0159%	0.1212%

For the experiment on Dataset 2, Table 4 shows the MAC value comparison for different methods on this dataset, and Table 5 provides a quantitative analysis of the inherent frequency errors. These tables clearly demonstrate the performance of different algorithms on Dataset 2, further validating the consistency and reliability of the proposed method across different types of data.

Table 4.

Comparison of MAC values for underdetermined modal parameter identification on Dataset 2.

Modal\Algorithm	Tucker	PARAFAC2	Bayes-CPD	APCA-BTD	APCA-CPD
1st	0.9710	0.9999	0.9827	0.9993	0.9999
2nd	0.9751	0.9658	0.9644	0.8826	0.9990
3rd	0.9250	0.9673	0.8870	0.9712	0.9997
4th	0.8981	0.9257	0.9160	0.7641	0.9987
5th	0.7809	0.9430	0.7697	0.9623	0.9505

Table 5.

Relative error in inherent frequencies for Dataset 2.

Algorithm\Modal		1st	2nd	3rd	4th	5th
Theoretical value (Hz)		8.900	55.780	156.200	306.090	505.840
Tucker	Identification value	9.003	56.410	157.800	308.800	510.130
Tucker	Relative error	1.157%	1.129%	1.024%	0.885%	0.848%
PARAFAC2	Identification value	8.958	56.380	157.900	308.800	510.100
PARAFAC2	Relative error	0.652%	1.076%	1.088%	0.885%	0.842%
Bayes-CPD	Identification value	8.981	56.490	157.870	308.860	509.980
Bayes-CPD	Relative error	0.910%	1.273%	1.069%	0.905%	0.818%
APCA-BTD	Identification value	8.955	56.400	157.800	308.700	509.700
APCA-BTD	Relative error	0.618%	1.112%	1.024%	0.853%	0.763%
APCA-CPD	Identification value	8.952	56.380	157.700	308.700	509.700
APCA-CPD	Relative error	0.584%	1.076%	0.960%	0.853%	0.763%

In addition to the quantitative analysis, Figures 6 and 7 provide qualitative comparisons by comparing the modal shapes identified by the APCA-CPD method with theoretical modal shapes (for Datasets 1 and 2). This helps visually assess the accuracy of the algorithm in modal identification. Figure 6 shows the comparison of the APCA-CPD method’s modal shapes with theoretical modes for Dataset 1, while Figure 7 provides a similar analysis for Dataset 2, visually demonstrating the agreement between the proposed method and the theoretical modes. Among them, UCIMS in Figures 6 and 7 represent the Identified Mode Shapes under Underdetermined Conditions.

Figure 6.

Comparison of identified and theoretical mode shapes on Dataset 1.

Figure 7.

Comparison of identified and theoretical mode shapes on Dataset 2.

In conclusion, through both quantitative and qualitative comparisons, the results from the two datasets indicate that the APCA-CPD method proposed in this study outperforms other benchmark algorithms in terms of modal parameter identification accuracy, inherent frequency estimation, and modal shape matching, demonstrating its superior performance in addressing underdetermined problems.

4.4. Ablation experiment

In order to verify the effectiveness of the proposed method of dynamic tensor construction with autocorrelation information, an ablation experiment was designed to replace the dynamic tensor construction strategy with the traditional static tensor construction method, and analyze the impact of such changes on the model performance. The experiment helps to verify the effect of dynamic updating tensor structure on improving mode capture ability. Through ablation experiments, the proposed APCA-CPD and APCA-BTD methods are compared with the corresponding methods (CPD and BTD) for dynamic third-order tensor construction without APCA method. The data sets used in the ablation experiment are the same as those used in the comparison experiment: Datasets 1 and 2. MAC values are presented in Tables 6 and 7, while the relative error of natural frequencies is detailed in Tables 8 and 9. To visualize the relative error between the natural frequencies identified by the APCA-CPD method and theoretical values, line graphs in Figures 8 and 9 illustrate the results from Tables 8 and 9. These graphs clearly demonstrate that the proposed method outperforms traditional methods across the board.

Table 6.

Comparison of MAC values for ablation experiment on Dataset 1.

Modal\Algorithm	BTD	CPD	APCA-BTD	APCA-CPD
1st	0.9838	0.9874	0.9966	1.0000
2nd	0.9494	0.9718	0.9761	1.0000
3rd	0.9357	0.9518	0.9619	1.0000

Table 7.

Comparison of MAC values for ablation experiment on Dataset 2.

Modal\Algorithm	BTD	CPD	APCA-BTD	APCA-CPD
1st	0.9806	0.9811	0.9993	0.9999
2nd	0.8786	0.9841	0.8826	0.9997
3rd	0.9767	0.9590	0.9712	0.9990
4th	0.7272	0.9757	0.7641	0.9987
5th	0.8537	0.9453	0.9623	0.9505

Table 8.

Relative error of natural frequencies for ablation experiment on Dataset 1.

Algorithm\Modal		1st	2nd	3rd
Theoretical value (Hz)		2.2399	6.2760	9.0690
BTD	Identification value	2.2420	6.2850	9.0980
BTD	Relative error	0.0937%	0.1434%	0.3198%
CPD	Identification value	2.2420	6.2770	9.0820
CPD	Relative error	0.0937%	0.0159%	0.1433%
APCA-BTD	Identification value	2.2420	6.2770	9.0770
APCA-BTD	Relative error	0.0937%	0.0159%	0.0882%
APCA-CPD	Identification value	2.2410	6.2770	9.0800
APCA-CPD	Relative error	0.0491%	0.0159%	0.1212%

Table 9.

Relative error of natural frequencies for ablation experiment on Dataset 2.

Algorithm\Modal		1st	2nd	3rd	4th	5th
Theoretical value (Hz)		8.900	55.780	156.200	306.090	505.840
BTD	Identification value	8.955	56.420	157.800	308.800	509.900
BTD	Relative error	0.618%	1.147%	1.024%	0.885%	0.803%
CPD	Identification value	8.952	56.380	157.900	308.700	509.500
CPD	Relative error	0.584%	1.076%	1.088%	0.853%	0.724%
APCA-BTD	Identification value	8.955	56.400	157.800	308.700	509.700
APCA-BTD	Relative error	0.618%	1.112%	1.024%	0.853%	0.763%
APCA-CPD	Identification value	8.952	56.380	157.700	308.700	509.700
APCA-CPD	Relative error	0.584%	1.076%	0.960%	0.853%	0.763%

Figure 8.

Line graph of natural frequency errors for Dataset 1.

Figure 9.

Line graph of natural frequency errors for Dataset 2.

The results of ablation experiments show that the autocorrelation optimization dynamic construction tensor contributes significantly to the model performance. In the absence of autocorrelation optimization, the modal recognition accuracy of the model will be reduced.

4.5. Noise addition experiment

To verify the robustness of the proposed method under noisy conditions, Gaussian white noise with different signal-to-noise ratios (SNR) was added to the vibration response signals, and the performance of the proposed method was compared with other benchmark methods for underdetermined modal parameter identification. The experimental results indicate that, even with varying levels of noise, the proposed method consistently outperforms the compared methods across both datasets. Specifically, Tables 10 and 11 present the comparison of MAC values and relative errors in natural frequencies for Dataset 1 under different SNR conditions, showing that the proposed method achieves superior accuracy and stability in modal parameter identification.

Table 10.

Comparison of MAC values under different SNR conditions (Dataset 1).

Noise	Method	1st	2nd	3rd
SNR = 20 dB	Tucker	0.8052	0.7527	0.6606
	PARAFAC2	0.8769	0.7920	0.6500
	Bayes-CPD	0.8607	0.7669	0.7323
	APCA-BTD	0.9720	0.9477	0.9220
	APCA-CPD	1.0000	1.0000	1.0000
SNR = 15 dB	Tucker	0.6357	0.6597	0.6246
	PARAFAC2	0.6870	0.6457	0.6304
	Bayes-CPD	0.7516	0.6696	0.6048
	APCA-BTD	0.7259	0.7640	0.8222
	APCA-CPD	1.0000	1.0000	0.9998

Table 11.

Comparison of relative errors in natural frequencies under different SNR conditions (Dataset 1).

Noise	Method		1st	2nd	3rd
Theoretical value (Hz)			2.2399	6.2760	9.0690
SNR = 20 dB	Tucker	Identification value	2.2420	6.2830	9.0800
	Tucker	Relative error	0.0937%	0.1115%	0.1212%
	PARAFAC2	Identification value	2.2420	6.2770	9.1850
	PARAFAC2	Relative error	0.0937%	0.0159%	1.2790%
	Bayes-CPD	Identification value	2.2440	6.2790	9.1150
	Bayes-CPD	Relative error	0.1830%	0.0478%	0.5072%
	APCA-BTD	Identification value	2.2420	6.2770	9.0700
	APCA-BTD	Relative error	0.0937%	0.0159%	0.0882%
	APCA-CPD	Identification value	2.2410	6.2770	9.0800
	APCA-CPD	Relative error	0.0491%	0.0159%	0.1212%
SNR = 15 dB	Tucker	Identification value	2.2420	6.2900	9.3800
	Tucker	Relative error	0.0937%	0.2231%	3.4293%
	PARAFAC2	Identification value	2.2420	6.3180	9.2611
	PARAFAC2	Relative error	0.0937%	0.6692%	2.1182%
	Bayes-CPD	Identification value	2.2440	6.3500	9.3500
	Bayes-CPD	Relative error	0.1830%	1.1790%	3.0985%
	APCA-BTD	Identification value	2.2420	6.2770	9.0800
	APCA-BTD	Relative error	0.0937%	0.0159%	0.1212%
	APCA-CPD	Identification value	2.2410	6.2770	9.0800
	APCA-CPD	Relative error	0.0491%	0.0159%	0.1212%

For Dataset 2, Tables 12 and 13 compare the MAC values and relative frequency errors under similar noisy conditions, further validating the significant advantages of the proposed method in complex noise environments. These experiments fully demonstrate the robustness and superior performance of the proposed method, which maintains high identification accuracy even under significant noise interference. By observing Tables 10 and 12, it can be concluded that the proposed APCA-CPD method achieves the highest MAC values (close to 1.000) under both 20 dB and 15 dB SNR conditions, demonstrating exceptional modal identification accuracy in noisy environments. In contrast, the accuracy of Tucker decomposition and PARAFAC2 decreases significantly under low SNR conditions, highlighting the robustness of APCA-CPD. Similarly, by observing Tables 11 and 13, it is evident that the APCA-CPD method exhibits the lowest relative errors across all modes, outperforming other methods under both high and low SNR conditions, showcasing its superior robustness and accuracy in noisy environments.

Table 12.

Comparison of MAC values under different SNR conditions (Dataset 2).

Noise	Method	1st	2nd	3rd	4th	5th
SNR = 20 dB	Tucker	0.9539	0.9139	0.8940	0.8531	0.7443
	PARAFAC2	0.8399	0.8006	0.8379	0.8746	0.8079
	Bayes-CPD	0.9163	0.8101	0.8628	0.8582	0.7377
	APCA-BTD	0.9568	0.8177	0.9296	0.7249	0.8578
	APCA-CPD	0.9999	0.9987	0.9989	0.9987	0.9505
SNR = 15 dB	Tucker	0.8612	0.8360	0.8734	0.6469	0.7050
	PARAFAC2	0.7155	0.7691	0.7147	0.7258	0.7539
	Bayes-CPD	0.7575	0.7768	0.7406	0.7063	0.6942
	APCA-BTD	0.9392	0.8092	0.8707	0.7075	0.8235
	APCA-CPD	0.9999	0.9987	0.9989	0.9987	0.9407

Table 13.

Comparison of relative errors in natural frequencies under different SNR conditions (Dataset 2).

Noise	Method		1st	2nd	3rd	4th	5th
Theoretical value (Hz)			8.900	55.780	156.200	306.090	505.840
SNR = 20 dB	Tucker	Identification value	9.003	56.485	157.876	308.863	510.176
	Tucker	Relative error	1.157%	1.264%	1.073%	0.906%	0.857%
	PARAFAC2	Identification value	8.958	56.783	157.942	308.879	510.181
	PARAFAC2	Relative error	0.652%	1.798%	1.115%	0.911%	0.858%
	Bayes-CPD	Identification value	8.965	56.896	157.883	308.945	510.136
	Bayes-CPD	Relative error	0.730%	2.001%	1.077%	0.933%	0.849%
	APCA-BTD	Identification value	8.955	56.446	157.862	308.762	509.910
	APCA-BTD	Relative error	0.618%	1.194%	1.064%	0.873%	0.805%
	APCA-CPD	Identification value	8.955	56.416	157.860	308.750	509.332
	APCA-CPD	Relative error	0.618%	1.140%	1.063%	0.869%	0.690%
SNR = 15 dB	Tucker	Identification value	9.008	56.985	158.176	309.151	510.996
	Tucker	Relative error	1.213%	2.160%	1.265%	1.000%	1.019%
	PARAFAC2	Identification value	8.962	56.879	157.976	308.969	510.778
	PARAFAC2	Relative error	0.697%	1.970%	1.137%	0.941%	0.976%
	Bayes-CPD	Identification value	9.057	56.945	158.031	309.186	510.760
	Bayes-CPD	Relative error	1.764%	2.089%	1.172%	1.011%	0.973%
	APCA-BTD	Identification value	8.955	56.465	157.878	308.839	510.227
	APCA-BTD	Relative error	0.618%	1.228%	1.074%	0.898%	0.867%
	APCA-CPD	Identification value	8.955	56.422	157.865	308.766	509.350
	APCA-CPD	Relative error	0.618%	1.151%	1.066%	0.874%	0.694%

5. Discussion

The experimental results from the comparative analysis, ablation studies, and noise-robustness tests comprehensively validate the effectiveness and advantages of the proposed APCA-CPD method. In comparison to benchmark algorithms such as Tucker decomposition, PARAFAC2, APCA-BTD, and Bayesian CP decomposition, the APCA-CPD method consistently demonstrated superior performance in identifying modal parameters under underdetermined conditions. Specifically, the results from Tables 2 –5 highlight its higher MAC values and lower relative errors in natural frequencies across two datasets, showcasing its robustness and reliability in both parameter estimation and modal shape matching. The ablation experiments further revealed that the incorporation of autocorrelation information and dynamic tensor construction significantly enhances the model’s performance. By comparing dynamic and static tensor construction strategies, it was observed that dynamic tensor construction improves the model’s ability to capture modes more accurately, as evidenced by the results in Tables 6 –9. Without autocorrelation optimization, the recognition accuracy of the model declines, underscoring the critical role of this feature in handling underdetermined systems. Moreover, noise robustness experiments using Gaussian white noise with different SNRs (Tables 10 –13) further validate the method’s effectiveness in noisy environments. The APCA-CPD method maintained near-perfect MAC values and the lowest relative errors even at low SNR conditions (15 dB), whereas traditional methods exhibited significant performance degradation. These findings emphasize the robustness and adaptability of the proposed method to real-world noisy scenarios. In summary, the APCA-CPD method not only addresses the limitations of existing algorithms but also sets a new standard for underdetermined modal parameter identification, offering exceptional accuracy, robustness, and versatility across various datasets and noise levels.

6. Conclusion

This paper presents a method for identifying the modal parameters of linear time-invariant structures, using an approach based on autocorrelation-optimized principal component CP decomposition. In this paper, the basic relation between the identification of operational modal parameters and tensor decomposition is established, and a new method of third-order tensor construction combining time series analysis and multidimensional data processing is introduced. This approach optimizes tensor construction according to the data’s statistical characteristics, making it particularly effective for extracting modal information from multichannel data under underdetermined conditions (where sensor numbers are limited but data richness is high). The method’s effectiveness is validated through numerical simulations and experiments, with results showing stable and accurate identification even in noisy, underdetermined settings.

Future research will focus on developing more efficient ways to determine the tensor decomposition rank R, which significantly impacts identification results, especially under dense modal conditions. Currently, rank selection relies on experimental trial and error, highlighting the need for optimized approaches to choose the best tensor decomposition rank for robust modal parameter identification.

Footnotes

Author contributions

Haonan Chen: Conceptualization, Methodology, Formal analysis, Writing—Original Draft.

Cheng Wang: Data curation, Resources, Writing—Review & Editing, Supervision.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by National Natural Science Foundation of China (Grant No.12372029) and research was supported by Xiamen City Construction Science and Technology Program for the Development and Application of Smart Construction Site Remote Online Real-Time Safety Monitoring System and this work is supported by The Major Science and Technology Special Project of Fujian Province (grant nos.2024HZ022013).

ORCID iD

Haonan Chen

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