Shock and Vibration

Volume 2019, Article ID 9602535, 11 pages

https://doi.org/10.1155/2019/9602535

## Short Data-Based Output-Only Identification for Time-Varying Systems with Fast Dynamic Evolution

^{1}School of Mechanical Engineering, Tianjin University, Tianjin 300350, China^{2}Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin 300350, China

Correspondence should be addressed to Qian Ding; nc.ude.ujt@gnidq

Received 13 March 2019; Revised 23 April 2019; Accepted 9 May 2019; Published 2 June 2019

Academic Editor: Itzhak Green

Copyright © 2019 Zhi-Sai Ma and Qian Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Many engineering systems change appreciably over a relatively short time interval due to their fast evolution in the dynamics. Time-varying (TV) system’s ambient excitation is usually difficult to measure under operating conditions, and its dynamics have to be determined without measuring the excitation. Therefore, short data-based output-only identification for TV systems with fast dynamic evolution is considered in this paper. Deterministic parameter evolution methods are known to track fast dynamic evolution by postulating TV model parameters as deterministic functions of time and selecting proper functional subspaces. However, these methods require a significant number of parameters to represent complicated time-dependencies and dynamics characterized by larger numbers of degrees-of-freedom. In such cases, the ordinary least squares estimation may lead to less accurate or even unreliable estimates. A ridge regression-based deterministic parameter evolution method is proposed to overcome ill-posed problems via regularization and subsequently assessed through numerical and experimental validation. Comparative results confirm the advantages of the proposed method in terms of achievable natural frequency and power spectral density tracking, accuracy, and resolution of TV systems with fast dynamic evolution, when the response data length is relatively short.

#### 1. Introduction

The need for time-varying (TV) system characterization is pervasive throughout the aerospace, mechanical, transportation, and manufacturing engineering practice. It is usually difficult to build the explicit dynamic model of a practical system via mechanism analysis; hence, TV system identification in a way that takes time variation explicitly into account is receiving more and more attention [1–4]. With the development of engineering applications, systems with fast varying dynamics are being widely used and their intrinsic nonstationary characteristics are increasingly inevitable. On the one hand, these systems may change appreciably over a relatively short time interval due to their fast evolution in the dynamics. On the other hand, the dynamic characteristics of TV systems may not fit into a laboratory and controlled testing may not be feasible under operating conditions. In other words, TV system’s ambient excitation is usually difficult to measure under operating conditions and its dynamics have to be determined without measuring the excitation. Therefore, short data-based output-only identification for TV systems is considered in this work in order to capture their fast varying dynamics.

Identification methods for TV systems are generally classified under the umbrella of time-frequency methods [1]. Most frequency-domain methods employ time-frequency analysis (e.g., the short time Fourier transform, Cohen’s class, wavelet-based methods, the Hilbert–Huang transform, etc.), which are based on nonparametric representations of the nonstationary signal as a simultaneous function of time and frequency. By contrast, time-domain methods employ parametric representations (e.g., time-dependent autoregressive moving average (TARMA) models), which offer a number of potential advantages, such as improved accuracy, resolution, and tracking of the TV dynamics. TARMA model-based methods may be further divided into three classes according to the type of “structure” imposed upon the evolution of the TV model parameters: unstructured parameter evolution (UPE), stochastic parameter evolution (SPE), and deterministic parameter evolution (DPE) methods [3]. The DPE methods are mainly based on explicit models of parameter evolution through projecting the parameter trajectory onto specific functional subspaces defined by deterministic basis functions (such as trigonometric, Legendre, Chebyshev, wavelets, B-splines, moving Kriging shape functions, radial basis functions, and others). By postulating TV model parameters as deterministic functions of time, the TARMA model can be transformed into a functional series (FS) TARMA model with time-invariant parameter matrix. Therefore, DPE methods are known to track fast dynamic evolution by selecting proper functional subspaces over their UPE and SPE counterparts [2–4].

DPE methods have played an important role in the development of TV system identification and nonstationary signal processing. In 1970, Rao [5] approximated AR coefficients by truncated Taylor series expansion and introduced the idea of representing TV model parameters as polynomials of time. Thereafter, Mendel [6] summarized the nonstationary identification methods and classified them into UPE, SPE, and DPE, which are still in use today. Kozin [7] approximated AR coefficients by a linear combination of Legendre basis functions and applied the FS-TAR model to earthquake ground motion analysis. In 1983, Grenier [8] extended the FS-TAR model to the FS-TARMA case by examining the existence and uniqueness of a FS-TARMA model representation for a nonstationary system, which was successfully applied to speech signal modeling [9–12]. Gersh and Kitagawa [13] extended the univariate FS-TAR model to the multivariate/vector case. Niedzwiecki [14, 15] defined the idea of expanding TV model parameters onto linear combinations of basis functions as “functional series modeling” and proposed a recursive least squares for the estimation of FS-TAR model parameters. Tsatsanis and Giannakis [16, 17] proposed FS-TARMA models with exogenous input and FS time-dependent finite impulse response models and considered the issue of model structure selection. In 2000, Niedzwiecki [2] summarized the DPE methods and investigated the time and frequency characteristics of basis function estimators.

In 2006, Poulimenos and Fassois [3] systematically reviewed the UPE, SPE, and DPE methods in terms of achievable model parsimony, model parameter estimation accuracy, tracking capability of TV dynamics, and computational simplicity. Spiridonakos and Fassois extended the FS-TAR model to the FS-TARMA case [18] and proposed the adaptable FS-TARMA model characterized by basis functions that are adaptable to the signal being modeled [19]. In 2014, Spiridonakos and Fassois [4] reviewed the conventional and adaptable FS-TARMA model in terms of parameter estimation, model structure selection, and model validation and outlined the developments of the DPE methods. Recently, Yang et al. [20] have proposed an adaptable FS-TARMA modeling method by selecting moving Kriging shape functions as basis functions. Zhou et al. [21] have proposed a least squares support vector machine-based FS-TAR modeling method by selecting compactly supported radial basis functions. Bertha and Golinval [22] have focused on the identification of TV mode shapes during the FS-TARMA modeling. Li et al. [23] have focused on the problem of extracting the real modal parameters from computational ones during the adaptable FS-TAR modeling. Ma et al. [24, 25] have focused on the recursive identification of TV systems by introducing the idea of kernel methods into the FS-TARMA modeling.

Despite the foregoing progress that has been achieved so far, there are still many issues that are open and require work, for example, the feasibility of TV system identification under a relatively limited data length. DPE methods can achieve better model parsimony over their UPE and SPE counterparts, yet they still require a significant number of parameters to represent complicated time-dependencies and dynamics characterized by larger numbers of degrees-of-freedom [3]. Therefore, the aim of the paper is two-fold: (1) the existing least squares-based FS-TAR modeling is reviewed to reveal the problem of TV system identification under a relatively limited data length and (2) a ridge regression-based FS-TAR method is subsequently proposed to track fast dynamic evolution of the system being modeled by using short data.

The paper is organized as follows: the least squares-based FS-TAR modeling is presented in Section 2. Section 3 proposes a new ridge regression-based FS-TAR method to identify TV systems with fast dynamic evolution under limited data length. The proposed and existing methods are numerically and experimentally tested in Section 4 and Section 5, respectively. Finally, Section 6 gives the remarkable conclusions.

#### 2. LS-Based FS-TAR Model Parameter Estimation

##### 2.1. FS-TAR Model

A time-dependent autoregressive (TAR) model with designating its autoregressive (AR) order is given by [18]where designates normalized discrete time, the discrete-time nonstationary vibration signal being modeled, an unobservable uncorrelated innovations (residual) sequence with zero mean and TV nonsingular covariance matrix , and the time-dependent AR parameter matrix. normally independently distributed random variables with the indicated mean and covariance.

Any given function can be approximated by a set of selected basis functions with arbitrary accuracy, as long as a sufficient number of basis functions is used [26]. Therefore, a TV parameter can be approximated bywhere denotes a set of basis functions selected from a suitable functional subspace, the coefficient of projection, the subspace dimensionality, and the index the specific basis functions of the selected functional subspace that are included in the basis.

Therefore, the time-dependent AR parameter matrix can be expressed as [3, 4, 18, 20]

The TAR model can be rewritten into the following FS-TAR model by substituting equation (3) into equation (1):withbeing the time-invariant parameter matrix, andthe corresponding regression vector, where is the Kronecker product [27], is given by

Evidently, the time-varying TAR model of equation (1) is transformed into a time-invariant FS-TAR model by expanding the time-dependent model parameters onto the selected functional subspaces. As given by equation (5), the FS-TAR model is fully described by parameters (coefficients of projection).

##### 2.2. Least Squares Estimation

Suppose the length of measured response data is , the matrix equation can be obtained by assembling each equation of form (4) for , as follows:with , , , and .

The parameter matrix can be estimated by using the well-known least-squares estimator through minimizing the sum of squared residuals, as follows:where denotes the Moore–Penrose pseudo inverse. The time-dependent AR parameter matrix can be subsequently estimated by using equation (3), after substituting by the obtained estimate .

However, if measured response data length is short and the inequality holds, there are fewer equations than unknown parameters and more than one satisfies equation (8). In other words, the solution is not unique and the problem is ill posed. In such cases, the ordinary least squares estimation is sensitive to errors and may lead to unreliable estimates. From the standpoint of the FS-TAR modeling, the data can be considered as “short” if the sample length () is smaller than the number of unknown parameters of the identified model ().

#### 3. RR-Based FS-TAR Model Parameter Estimation

##### 3.1. Ridge Regression Estimation

Ridge regression, also known as Tikhonov regularization, is a commonly used method of regularization of ill-posed problems [28, 29]. Given the regularization parameter , the least squares cost function is defined as

By using the ridge regression, we have the solution to equation (10) aswhere and .

By using the matrix inversion lemma, we have

Obviously, the computational complexity of the matrix inversion in equations (11) and (12) is, respectively, and . In order to avoid the matrix inversion, we further define and as

Then, we have

The update rule for the inverse of the above growing matrix iswith and .

By combining equations (12), (13), and (16), we havewith the final estimate . Similarly, the time-dependent AR parameter matrix can be subsequently estimated by using equation (3), after substituting by the obtained estimate .

##### 3.2. FS-TAR Model-Based Structural Dynamics

Once have been estimated, the system’s “frozen-time” poles can be obtained by solving the following general eigenvalue problem [20]:where designates the identity matrix of order and and , respectively, the poles and eigenvector of at time instant with the mode shape . The matrix is constructed from as

The “frozen-time” natural frequencies of the system can be computed bywhere designates the absolute value and the sampling period.

The response “frozen-time” power spectral density (PSD) for each time instant can be obtained aswhere designates frequency in , the imaginary unit, the Hermitian transpose, and , with being the window length.

#### 4. Numerical Validation

The performance characteristics of the proposed RR-FS-TAR method, along with the LS-FS-TAR method, are examined via Monte Carlo study focused on the identification of a TV three degrees-of-freedom structural system subject to a Gaussian, zero-mean, and uncorrelated excitation , as shown in Figure 1 [3, 30]. Numerical values of the system parameters are given in Table 1.