Shock and Vibration

Volume 2018 (2018), Article ID 7691721, 15 pages

https://doi.org/10.1155/2018/7691721

## A Novel Coupled State/Input/Parameter Identification Method for Linear Structural Systems

^{1}School of Vehicle and Transportation Engineering, Nantong Vocational University, Qingnian Middle Road No. 89, Nantong, China^{2}State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China^{3}School of Mechanical Engineering, Nantong Vocational University, Qingnian Middle Road No. 89, Nantong, China

Correspondence should be addressed to Ting Wang

Received 18 August 2017; Revised 30 November 2017; Accepted 23 January 2018; Published 20 February 2018

Academic Editor: Tony Murmu

Copyright © 2018 Zhimin Wan et al. 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

In many engineering applications, unknown states, inputs, and parameters exist in the structures. However, most methods require one or two of these variables to be known in order to identify the other(s). Recently, the authors have proposed a method called EGDF for coupled state/input/parameter identification for nonlinear system in state space. However, the EGDF method based solely on acceleration measurements is found to be unstable, which can cause the drift of the identified inputs and displacements. Although some regularization methods can be adopted for solving the problem, they are not suitable for joint input-state identification in real time. In this paper, a strategy of data fusion of displacement and acceleration measurements is used to avoid the low-frequency drift in the identified inputs and structural displacements for linear structural systems. Two numerical examples about a plane truss and a single-stage isolation system are conducted to verify the effectiveness of the proposed modified EGDF algorithm.

#### 1. Introduction

The state of a system is vital to structural health monitoring in mechanical engineering and civil engineering. The topic of identifying the state has been studied intensively for many decades. Various state estimators [1–7] for structural systems behaving both linearly and nonlinearly have been proposed, including the well-known Kalman filter (KF) [8] for linear structural systems.

However, only adopting KF to identify the structural state is often insufficient due to the uncertain structural parameters. The extended Kalman filter (EKF) [9], used for slightly nonlinear system, was proposed by linearizing the dynamic model. EKF has been one of the most widely used tools for joint input/parameter identification in structural dynamics, and it has been adopted in many applications, such as damage identification [10, 11], parameter identification [12, 13], and model updating [14].

In all of the above EKF applications, the input forces are assumed to be known. Unfortunately, the measurement of the input forces on a structure is not straightforward, because the introduction of dedicated force cells requires alteration to the structure to locate the sensor in the force path, which is unwanted and unpractical. The indirect measurement of the input forces, like calibrated strain gauges, requires a good knowledge of the structural parameters, which might be unknown as well. On the other hand, input forces together with the system states can be identified through a variety of KF approaches and acceleration measurements, which have gained increasing attention in recent years. Gillijns and De Moor [15] proposed a recursive optimal filter of joint state/input estimation for linear systems with direct transmission, which was originally presented for optimal control applications. The estimated input and state are optimal in a minimum-variance unbiased sense. Lourens et al. [16] extended this filter proposed by Gillijns and De Moor (GDF) of joint input/response identification for structural systems based on reduced-order model and acceleration data from a limited number of sensors. Maes et al. [17, 18] presented an extension of the GDF algorithm for joint input/state estimation in structural dynamics, accounting for the correlation between process noise and measurement noise. In [19], an augmented Kalman filter for joint state/force identification in structural dynamics was presented, in which the unknown forces and states were reformed into augmented states. And the KF was employed for identifying the novel augmented states. Naets et al. [20] adopted a similar algorithm for identifying the states and forces of multibody model. The difference is that EKF is employed for the joint state/input identification. In [21], the Popov-Belevitch-Hautus (PBH) criterion was used to evaluate the observation of the system for the augmented KF algorithm. It was shown that only using acceleration measurements inherently would bring unreliable results. To avoid this deficiency, the addition of dummy measurements on a position level was proposed. In [22, 23], a DKF (Dual KF) approach of joint state/input identification was proposed for linear state-space models. These methods in this paragraph associated with joint state/input identification are based on an accurate structural model in which there are no uncertain parameters. However, it is still impossible to obtain a completely accurate model due to the structural complexity and the manufacturing or measuring errors.

From the above literatures, it is known that the identification of forces and parameters cannot be separated for general structures. For improvement, several approaches have been developed for joint input/parameter identification. Kolmanovsky et al. [24] developed the coupled identification of an input and set-membership parameter while this algorithm requires fully known states, which is rarely the case in engineering applications. Yang et al. [25] proposed an EKF with unknown input forces, referred to as EKF-UI, for the coupled state/input/parameter identification for structural damage detection. Recently, Lei et al. [26] applied the least-squares estimation algorithm to deduce the EKF-UI algorithm again and pointed out that the analytical recursive solutions by the original EKF-UI were acquired by relatively complex mathematical derivations. Lei et al. [27] also applied the above simplified EKF-UI algorithm for nonlinear structural parameter identification. However, in the simplified EKF-UI algorithm, the prior probability density functions (PDFs) of states at time are taken as the posterior PDFs of states at time, which may cause the confusion of concepts in Bayesian framework. Another method called augmented discrete extended Kalman filter (A-DEKF) for the coupled state/input/parameter identification was proposed by Naets et al. [28]. A-DEKF is similar to the augmented KF algorithm [19] where a high dimensional augmented state vector was formed by the states, unknown forces, and uncertain parameters. Moreover, a suitable assumption on the statistics of the force is needed for the A-DEKF method. In [29], the same methodology is also applied to an offshore wind turbine in order to simultaneously estimate the hydrodynamic loading, states, and a stiffness-related system parameter. Recently, a DKF-UKF framework has been developed for the coupled identification topic in [30, 31]. A DKF is used to identify the input, and an UKF (Unscented KF) is adopted to identify both the states and the unknown parameters (the so-called augment states). However, the stability, observability, and convergence analysis of DKF-UKF are not mentioned. Also a prior assumption on the covariance of the input is needed in DKF, which strongly influences the quality of the estimates of the Bayesian filters. The authors also proposed a method based on the GDF algorithm for coupled state/input/parameter identification for nonlinear systems [32]. The EGDF (extended GDF) algorithm is developed for the weak nonlinear systems by the linearization idea of EKF. The uncertain parameters and states are considered as the augmented states to be identified. Thus, the novel state transmission and measurement equations become nonlinear, which are linearized by the first-order Taylor expansions. The proposed EGDF has the same structure of the standard GDF algorithm, including three steps: input identification, measurement update, and time update. The main difference between them is the sensitivity matrices of the two nonlinear state-space equations. However, it has been demonstrated that the conventional GDF approach based on limited number of acceleration measurements is inherently unstable which leads to the so-called spurious low-frequency drift in the estimates of the force and the structural displacement [22]. Although some regularization methods or postsignal processing schemes can be adopted to deal with the drift in the identified results [33–36], they prohibit the real-time identification of coupled state/input/parameter.

In this paper, a strategy of data fusion of partially measured displacement and acceleration responses is adopted to avoid the so-called drifts in the identified displacement and force in the conventional GDF approach, since displacement and acceleration measurements contain low and high frequencies vibration characteristics, respectively.

This paper proceeds as follows. In Section 2, the standard GDF algorithm for joint state/input identification is presented. In Section 3, the nonlinear identification model of the coupled state/input/parameter is firstly built, and then the EGDF method is demonstrated, including data fusion of partially measured acceleration and displacement responses. Section 4 conducts two numerical examples to demonstrate the effectiveness of the modified EGDF method. Finally, the conclusion is drawn.

#### 2. Joint State/Input Identification

##### 2.1. Discrete-Time State-Space Model of Structural Dynamics

The general equation of motion of a damped structure with DOFs can be expressed aswhere , , and are the mass, damping, and stiffness matrices of the structure, respectively; , , and are, respectively, the nodal acceleration, velocity, and displacement vectors of the structure; is the force vector and is the influence matrix associated with the input .

The second-order equation of motion (1) can be transformed into a first-order continuous-time equation in state space aswhere is the state vector, and the system matrices together with are, respectively, defined as

Assuming that only acceleration responses are measured, the measurement equation can be changed into the following state-space form:with the output influence matrix and direct transmission matrix defined aswhere represents the selection matrix of acceleration measurements.

The time step size is denoted as the symbol ; thus the continuous-time state-space model of (2) and (4) can be transformed into the discrete-time form using a sampling rate of 1/ aswhere

##### 2.2. The GDF Algorithm for Joint State-Input Identification

Adding the system and measurement noise to the linear structural system, the system equations can be rewritten as The system noise vector and measurement noise vector are assumed to be mutually uncorrelated, zero-mean, white random signal with known covariance matrices, and , respectively. Results are easily generalized to the case where and are correlated [37]. Results can also be easily generalized to systems with both known and unknown input. It is obviously seen that (8) represent the linear system of structural dynamics with direct feedthrough. In the field of system control, Gillijns and De Moor proposed an optimal recursive filter of the direct feedthrough case for joint input/state identification [15]. In this filter, a state estimate is defined as an estimate of given and its error covariance matrix expressed as . An initial unbiased state estimate and its covariance matrix are assumed known.

The GDF algorithm computes the unknown force and state in a recursive procedure including three steps: the step of the input identification, the measurement update, and the time update. The GDF algorithm is listed in Algorithm 1.