A Novel Coupled State/Load/Parameter Identification Method Based on the Improved Particle Filter Algorithm

There have been increasing research studies focusing on the identification of structural load with uncertain model in recent years. This paper presents a novel approach based on the improved Particle Filter (PF) algorithm for the nonlinear structural system. The main strategy of the approach is the fusion of the weighted least-square algorithm and the conventional PF algorithm. The weighted least-square algorithm is derived for the estimation of the unknown loads, and the PF algorithm is used for the identification of the augmented states which includes the structural displacement, velocity and unknown parameters. Additionally, the parallel algorithm is adopted for the improvement of computing speed, and a resampling step is also used for alleviating the degeneracy of the particles to improve the accuracy. A numerical example modelling as a four-story hysteretic shear-beam building is studied to validate the capability of the presented approach.


Introduction
The structural dynamic load identification is the second type of inverse-problem in the field of structural dynamics.The load is usually obtained by the measured signal and structural system, and the main methods are respectively the time-domain methods [1][2] and the frequency-domain methods [3][4].The time-domain methods are more suitable for the estimation of the non-stationary and instantaneous load compared to the frequency-domain method, and are widely concerned and developed.Several algorithms are proposed for solving the ill-conditioned difficulty in inverse problem, including the SVD algorithm [5], the direct regularization [6], the sparse regularization [7], the iterate regularization [8] and the basis-function decomposition [9] et al.However, the structural system is always with some uncertain parameters inevitably in mechanical engineering and civil engineering due to the influence of manufacturing, measurement, wear and some other factors.Therefore, the traditional time-domain methods are not suitable for the load identification of the models with the uncertain system parameters.In recent two decades, several types of algorithms have been proposed for the structural dynamic load identification with the unknown parameters, such as the interval perturbation algorithm [10], probability statistics algorithm [11], Bayesian regularization algorithm [12], iterate least square algorithm [13], dynamic response sensitivity-based algorithm [14] and Kalman-based algorithm [15].Among these algorithms, the Kalman-based type algorithms which can identify structural states with uncertain parameters are presented for many research fields such as the first inverse problem and second inverse problem in structural dynamics, structural health monitoring and structural damage identification.According to research literature, they mainly include the EKF-UI [16][17], the dual Kalman Filter (DKF) [18][19], the A-DEKF [20], the MEKF-UI [21] and the EGDF algorithm proposed by the authors [22].The same disadvantage among these algorithms is the so-called spurious low-frequency drift in the estimated displacement and load when only partial acceleration signal are measured for identification.The drift is caused by acceleration's insensitivity to any quasi-static component in the input external load.Some regularization methods and postsignal processing schemes have been proposed for dealing with the drift in the identified load and displacement [23].However, they cannot be useful for the real-time identification.Furthermore, the data fusion of the structural strain and acceleration signal, displacement and acceleration are presented to alleviate the low-frequency problem respectively [21][22]24].In addition, these algorithms are proposed for the linear system and the weak non-linear system, which are not suitable for strong nonlinear system.Song [25] proposed a dual load/state identification method based on the unscented KF algorithm for non-linear structural system, i.e. the unscented transformation was used for no-linear approximation, and the GDF algorithm [26] was adopted for identification.Particle Filter is the algorithm of probability statistic, which is suitable for non-linear/non-Gussian system.In Ref [27], a new approach for the joint estimation of state/load/parameter was presented.The undetermined parameters were added to the structural states (displacement and velocity), then the extensive ones were formed as the augmented states for recognition.The PF algorithm was used to identify the augmented states, and the least-squares estimation algorithm was adopted to obtain the unknown loads.
In this paper, a new method of the coupled state/load/parameter identification is presented based on Ref [27].The identified load is presented using a more strict mathematical derivation, which is minimum variance unbiased estimation.Then the augmented state is identified by the PF algorithm.Additionally, to resolve the spurious low-frequency drift problem, the data fusion strategy of structural displacement and acceleration is adopted, and the parallel resample is used for mitigating particle degeneracy.Numerical study of a typical nonlinear system is studied to illustrate that the presented method is efficient for the coupled state/load/parameter identification.

Nonlinear Structural System Model
Assuming a nonlinear structure system has n degrees of freedom, the general equation of motion of the nonlinear structure can be expressed as follows  …,d) being the unknown parameter of the nonlinear structures; and   ( ), ( ), tt F p p α represents an internal load vector, which is a nonlinear function associated to the above () t p () t p and α .The mass matrix M is assumed as being known in this paper, because the mass can be obtained easily by the structural geometry and material information.
The augmented state vector z(t) is represented as , which includes the traditional state (consisting of displacement and velocity) and unknown structural parameters.The structural parameters are defined to be invariable in this paper, so that one can obtain the equation   = α 0 .The above equation ( 1) can be transformed into another form called as the augmented state transmission equation as follows is the continuous-time form of the nonlinear augmented state transmission function.
The observation equation can be also obtained as the augmented state-space form as follows ( ) represents the nonlinear function associated to the augment state () t z .
(1) If the measured signal is only the acceleration, then equation (3) can be transformed into in which Ha is the mn  measured acceleration influence matrix, and ( ) (2) If measured signal are the acceleration and displacement signals, then equation (3) can be transformed into where Hd is the lr  measured displacement influence matrix, and Considering the process noise, the above continuous-time augmented state-space equations can be rewritten as the following discrete-time schemes where subscript symbol k represents the kth time instant; the symbols sk and vk are the system noise and measurement noise vector, respectively.And the two noise vectors are defined to be mutually uncorrelated, which are both the zero-mean white random signals with the known matrices as Gk and Rk.

Load Identification
In this section, the load identification is deduced by the minimum-variance unbiased estimation.Adopte the following expression as where ( ) represents the mean.Therefore, it follows from observation equation ( 11) that, where ek is written as Obviously, if the augmented state zk is unbiased, then one can obtain the equation ( ) equation (14).From equation ( 13), one can obtain that ( ) ( ) Obviously, this indicates that an unbiased estimate of the undetermined loads uk can be computed from the expression k y .Assume that the load of unbiased estimation can be computed as ( ) From equation ( 15), it requires kk = J D I , the identity matrix.The matrix k D is full column rank, with the equation rank(Dk) = r.Moreover, the covariance of the error ek in equation ( 14) requires to be examined due to obtaining a minimum-variance estimate, which is written as in which the symbol c represents a positive real number.Obviously, equation ( 13) does not satisfy homoscedasticity, which shows that the error ek does not have uniform variance.Then, it content with the Gauss-Markov theorem [28] that the load in equation ( 15) is not necessarily minimum variance.
To obtain the unknown load, the optimal matrix Jk must be determined according to equation (15).Since to equation (13), one can obtain the following equation Now the covariance of the above disturbance is as  ) ( ) ( ) With the equation = e L L P , then Equation ( 18) can be simplified as ( ) ( ) where ( ) ( )

The Improved PF Algorithm
The novel improved PF algorithm is presented using the load identification and the traditional PF algorithm, which includes four steps as: initialization, load identification, measurement update and time update.
Step 1: Initialization Assume that the two matrices | ˆkk z and ˆk u are respectively the posterior unbiased estimate values of k u and k z based on the observed vector value(y0,y1,y2,…,yk).The augmented state variance matrix is defined as the expression , in which L is the filter number of PF, and N is the particle number of each PF process.
Step 2: Load identification In the load identification step, six equations are as following ( ) ( ) where matrix Dk is constant, i.e.Dk = D.
Step 3: Measurement update The importance weight needs to be updated, and the proposal density uses the ( ) In the traditional scheme of the basic PF algorithm, the so-called particle degeneracy phenomenon is very difficult to be avoided when only importance weight is used [27].In order to alleviate the degeneracy of the particles, the strategy of a resampling step is adopted.It is to abandon the particles of small weight and duplicate the particles of high weight.In this paper, the coefficient of variation (simplified as COV) of the importance weights is adopted, and the resampling step is computed only when the COV value exceeds the prescribed threshold, which indicates that the computed variability of the importance weight is large.The COV is written as .Otherwise, the resampling step is not executed, and the importance weight is normalized as , , , 1 The previous samples are storing as Then the unbiased minimum-variance of the augmented state is obtained as Step 4: Time update At time instant k+1, the particles samples are obtained from the following proposal probability distribution as Set k = k + 1 and repeat to do steps 2 to 4 until k = T.

Numerical Study
In this section, a four-story hysteretic shear-beam building with an unknown external load on the top floor of the structure is evaluated, which is shown as figure 1.
A hysteretic shear-frame building with undetermined external load acting at the 4 th floor.
The equations of structural dynamics are expressed as follows is the nonlinear function responding to the   ( ), ( ) tt F p p in equation ( 1); the symbol () t r represents the vector of hysteretic displacement of () i rt (i = 1,2,…,4) expressing the ith floor hysteretic restoring force and is modelled as the Bouc-Wen non-linear differential equation, which can be given as follows  represent the Bouc-Wen hysteretic parameters.The hysteretic force is hereditary, depending on the past history of deformation, and its description is very complicated.
Obviously, the shear-beam building structure is non-linear with the Bouc-Wen model.And also it is because that the structural hysteretic performance can indicate the development of structural damage under dynamic external load, so the structural damage can be found by the identification of hysteretic parameters.
In this example, the mass value of each floor is equal expressing as 1 6 0.9 . The external unknown load on the 4th floor is defined to be the non-Gaussian as the student's t distribution, and the corresponding degree of freedom is 5. Three acceleration response signals at the 1st, 2nd and 4th floors are the known signals, and the displacement response at the 3rd is also be measured for dealing with the low-frequency drift problem.3% RMS environment noise are contained in the measured signals.Two situations are considered for validation.One is for the joint state/load identification using the improved PF algorithm without structural parameters in the augmented state, and the other is for the identification of coupled state/load/parameter using the improved PF algorithm. are assumed to be known.The sampling time interval is 0.002 s, and the importance weight COV threshold is determined as 200%.the actual and identification values of the load is in figure 2. From this figure, it can be found that the two curves are almost overlap, which indicates that the load is identified accurately.The good agreement can be also seen in figure 3, which are the identification results of the displacement and velocity at the 2nd floor.
where S represents one of the three physical quantities as the load, displacement, and parameters.(1) Different particle number From Group (1) and Group (3), one can know that the numbers of the parallel algorithms are the equal.For this compared case, it can be known that when the more particles are adopted in each independent PF algorithm, the identified results are more accurate.The same conclusion can be also obtained from the comparison of the identified REs value of Group (2) and Group (4).
(2) Effect of parallel computation From Group (1) and Group (2), it is found that the parallel computation is useful for improving the identified precision.The same conclusion is found from the comparison of the identified Res of Group (3) and Group (4), and it can also be known that the parallel computation is more accurate when each independent PF algorithm has less particles.
The actual and identified values of the unknown hysteretic parameters from Group (2) are shown in figure 4. The good convergence results are presented.From the above numerical analysis, it can be illustrated that the proposed method based on the improved PF algorithm is efficient for the coupled identification.

Conclusions
In this paper, a novel approach for coupled state/load/parameter identification of nonlinear/non-Gaussian system based on the improved PF algorithm, which includes two steps.One is the load identification of unbiased minimum-variance estimate based on the weighted least-square algorithm.The other step is the augmented state identification based on the conventional PF algorithm, and the augmented state includes structural displacement, velocity and undetermined parameters.Additionally, the data fusion of displacement and acceleration is adopted for dealing with the low-frequency drift in identification results, and the parallel computations are also used to improve the accuracy of identified results and computing speed.Numerical study of a four-story hysteretic shear-beam building subject to unmeasured external non-Gaussian type load is conducted to evaluate the correctness of the presented method.The effects of the different particle number and parallel computation are demonstrated respectively.Two conditions are required as follows: the first one is the number of the measured signals should be larger than that of the undetermined excitation loads; and the second one is the acceleration measurement signals should be available at the locations in which the undetermined loads act.
L e I , which obviously satisfy homoscedasticity.According to the assumption that 1 kk − LD has full column rank, the unbiased minimum-variance load can be computing based on the Gauss-Markov theorem as follows

(
std represents the standard deviation, and mean is the mean of wk.Compute the COV value of , ij k w .If the COV value is larger than the setting threshold, the above particles resampling step is computed for i = 1,2,…,N with setting , ; the stiffness value of each story is equal expressing as 1 coefficient of each story is equal expressing as 1 2 3 4 1Ns / m c c c c = = = = ; the hysteretic parameters are unknown, and have the math relations as 1 2 3 4

Situation 1 :
The augmented state is r , and the external load is to be identified.The nonlinear hysteretic parameters j j

Figure 2 .
Figure 2. The actual and identified values of the exernal load u.

Figure 3 .Situation 2 :=-0. 12 
Figure 3. (a) The actual and identified displacement at the 2 nd floor.(b) The actual and identified velocity at the 2 nd floor.(c) The actual and identified hysteretic displacement at the 2 nd floor.

Figure 4 .
Figure 4.The actual and identified values of the unknown hysteretic parameters.

Table 1 .
The relative errors of the identified values by different cases.