2018 State of Art Review for Online System Identification Techniques in Control Systems Area

Introduction

In contempo years, an increasing attention is witnessed to confront the challenging problems of rubber, serviceability, reliability, chance and life-cycle direction, and functioning improvement of structures and infrastructure due to changing and more frequently occurring natural and man-made hazards, infrastructure crisis, and sustainability issues. These disturbances are dealt with innovative technologies to enhance structural functionality and prophylactic in various stages of research and development (Spencer, 2003; Altabey, 2017a). Several types of structures that employ control strategies for different awarding scopes can be institute in Constantinou et al. (1998), Soong and Spencer (2002), and Altabey (2014, 2017b,c); Altabey (2018). Proper modeling of inherent non-linearity in vast majority of structural systems plays an important part in agreement structural response under chancy loadings. System identification is an important approach in control strategy regarded as the interface between the mathematical earth of control theory and the existent earth of awarding and model abstractions (Zadeh, 1956; Ljung, 2010; Altabey, 2016, 2017d,e; Altabey and Noori, 2017a, 2018; Zhao et al., 2018), and it handles a wide range of system dynamics problem without the prior knowledge of bodily organisation physics. The schematic diagram of arrangement identification process is depicted in Figure ane.

Effigy 1. A diagram of system identification.

Hysteresis tin can be described as the hereditary and memory nature of a not-linear or inelastic organisation behavior where the restoring force is dependent on both instantaneous as well as past history of deformations. In general, nether cyclic loading, mechanical and structural systems are capable of dissipating considerable energy and they showroom observable hysteretic behavior with hysteresis loops. Each loop enclosing the expanse in the restoring strength vs. displacement curve depicts the energy dissipated over a consummate wheel resulting from internal friction inside the structural organisation.

Diverse empirical hysteresis models accept been proposed in the by few decades. A class of smoothly varying hysteresis models used in engineering fields are Bouc-Wen class of hysteresis models. Bouc suggested a smoothen and versatile hysteresis model for non-linear systems and hysteretic systems (Bouc, 1967; Wen, 1975, 1976, 1980, 1986, 1989; Park et al., 1986; Wen and Yeh, 1989; Ikhouane and Rodellar, 2007; Ikhouane et al., 2007; Ikhouane and Gomis-Bellmunt, 2008). Baber and Wen extended the Bouc model to take the degradation in forcefulness or stiffness of structural systems into account (Baber and Wen, 1980). Baber-Noori, and later on Noori, further extended the capabilities of Bouc-Wen model by including pinching beliefs and studied the response of these systems under random excitation (Noori, 1984; Baber and Noori, 1986). Baber-Noori and afterward Noori-Baber's work on integrating the pinching phenomenon in hysteretic behavior and extending Bouc-Wen-Baber (BWB) model was the first work in developing a shine hysteresis model capable of taking into account forcefulness and stiffness degradation as well as shear pinching phenomenon (Baber and Noori, 1985). BWBN was incorporated in structural design software, OpenSees developed at the University of California Berkeley (Hossain, 1995). A toolbox for calculating the parameters of BWBN hysteresis model using multi-objective optimization evolutionary algorithms was also developed by SourceForge, an Open Source community (Bouc Wen Baber Noori Model of Hysteresis, Source Forge). Foliente showed Bouc-Wen-Baber-Noori (BWBN) model could produce previously observed inelastic behavior of wood joints and structural systems using BWBN smoothen hysteresis model (Foliente, 1995; Zhao et al., 2017a,b; Noori et al., 2018). Deb et al. adult a toolbox that identifies structural parameters of Bouc–Wen–Baber–Noori hysteresis model through a noval multi-objective optimization evolutionary algorithms (MOBEAs) (Deb et al., 2002; Deb, 2013). Ortiz et al. analyzed and identified BWBN model via a multi-objective optimization algorithm (Ortiz et al., 2013). Peng et al. utilized BWBN model for identifying the parameters of a magneto-rheological damper and depicts its strength-lag phenomenon (Peng et al., 2014). Muller et al. investigated the awarding of BWBN in their work and conducted operation-based seismic design through a Search-Based Cost Optimization (Muller et al., 2012). Chan et al. made a prediction of the hysteretic behavior of passive control systems by applying BWBN in a nonlinear-autoregressive-exogenous model (Chan et al., 2015).

Traditional artificial neural networks technique shows its superiority in the identification, monitoring, and command of complicated and non-linear dynamic systems (Narendra and Parthasarathy, 1990; Masri et al., 1992; Lu and Basar, 1998; Abouelwafa et al., 2014; Altabey and Noori, 2017b). However, a priori knowledge of the characteristics of restoring force is necessary and important for traditional parametric organization identification approaches, while the non-parametric methods do non need information beforehand, lacking direct association between arrangement dynamics and system model. In order to overcome the limitations of conventional parametric and not-parametric approaches, an noval Intelligent Parameter Varying (IPV) method was proposed, which makes full apply of the embedded radial basis part networks to make an interpretation of the hysteretic and inelastic characteristics of restoring forces constitutively for a multi degree of freedom system. A scaled three story base excited structure was designed to experimentally verify the non-linearity and its associated hysteresis of a structure using a displacement controlled shaking table (Saadat et al., 2003, 2004a,b, 2007). Farther, a data-driven identification strategy for non-linear and hysteretic behaviors of steel wire strands was compared and verified using polynomial basis functions and neural networks. The results showed that neural networks were found more promising for the prediction of slightly pinched, hardening hysteresis, strongly pinched, hardening hysteresis, and classical quasi-linear softening hysteresis (Brewick et al., 2016). Genetic algorithms accept been used for organization identification of not-linear and hysteretic systems. The application of Existent Coded Genetic Algorithms (RCGA) was demonstrated and applied to fit curves of synthetic and experimentally obtained Bouc-Wen hysteresis loops for a sandwich composite cloth (Hornig and Flowers, 2005). Different real coded genetic algorithms and their related criteria for efficiently identifying not-linear systems are regards equally non-classical and optimized identification techniques (Monti et al., 2009). A Bayesian probabilistic framework was proposed to notice impairment of continuous monitored structures by incorporating load-dependent Ritz vectors equally an alternative to modal vectors (Sohn, 1998). A big trunk of work was conducted to track, gauge and identify structural parameters, system status and hysteretic and degrading behavior of structures using Kalman filters, extended Kalman filters and unscented Kalman filters (Jeen-Shang and Yigong, 1994; Yang et al., 2006; Wu and Smyth, 2007, 2008; Chatzi and Smyth, 2009; Chatzi et al., 2010; Lei and Jiang, 2011; Mu et al., 2013; Kontoroupi and Smyth, 2017; Erazo and Nagarajaiah, 2018). Traditional Markov chain Monte Carlo approach in conjunction with Bayesian updating method were applied for structural response predictions and operation reliability evaluation (Yuen and Katafygiotis, 2001; Zhang and Cho, 2001; Brook and Au, 2002). Later, a transitional Markov Concatenation Monte Carlo (TMCMC) approach was adult past designing optimized sampling strategy from a series of intermediate probability density functions (PDFs) that converge to the target PDF, thus avoiding sampling difficulties. The TMCMC theory and algorithm were verified and demonstrated through the performance of the developed sampling approach, different PDFs too as college dimensional issues (Ching and Chen, 2007; Muto, 2007; Muto and Brook, 2008; Worden and Hensman, 2012; Zheng and Yu, 2013; Behmanesh and Moaveni, 2014; Green, 2015; Dark-green et al., 2015; Ortiz et al., 2015).

It is a major barrier to successfully design hysteretic structures against deposition under severe circadian loading. Nearly structural systems dethrone with significant hysteresis, for example wood structures, dams, highways, reinforced concrete towers, steel bridges are disquisitional and key elements of our built surroundings. In spite of their obvious importance, and their huge rehabilitation and replacement costs, pattern, construction, and assay of the majority of these structures requires overly simplistic or in some cases flawed assumptions regarding hysteretic evolution. Development of a practical structural degrading identification approaches is much deserving. A comparative report of online and offline identification strategies for UAVs were discussed, and it is institute that online arroyo is more than adaptive to changes simply with lower prediction accuracy (Puttige and Anavatti, 2007). Therefore, offline learning is employed in this newspaper. Based on what was discussed in the introduction higher up, the main contributions of the research are to nowadays a iii story hysteretically degrading shear structure by incorporating BWBN slip lock hysteresis to represent the hysteretic restoring forces in this system. This BWBN model will be capable of producing all significant and prominent features of structural strength and stiffness degradations as well equally sideslip lock behavior, and acquit a comparative study using three system identification approaches including an Intelligent Parameter Varying Artificial Neural Network developed in an earlier research work past a group that involved one of the authors (a "grayness box" model that considers linear likewise equally non-linear parts of the dynamic organization), genetic algorithm optimization method, and a novel TMCMC statistical approach.

The comparative report of the aforementioned approaches for organisation identification and their application in a structural system using BWBN MODEL is an original work. To the best of the authors' knowledge such comparative study has not been reported in the literature.

Structural Arrangement Modeling and System Identification

BWBN Hysteresis Model

The model employed herein is an earlier version of BWBN hysteresis degradation model, which incorporates the previous smooth system degrading element past Bouc as modified by Baber and Wen in serial with a slip-lock element (a non-linear hardening jump) adult by Baber and Noori. Nether cyclic excitation, degradation manifests itself in the evolution of progressively varying hysteresis loops. A non-linear organization governed by Equation (1) is given with the incorporation of BWBN model.

$\begin{array}{cc}\dot{z}=\frac{A{\dot{x}}_1-\nu \left[\beta \right|{\dot{\mathrm{ten}}}_1\left|\right|z{|}^{n-\mathrm{i}}z+\gamma {\dot{x}}_1\left|z{|}^n\right]}{\eta }& (\mathrm{iii})\end{array}$

$\begin{array}{cc}{\dot{x}}_2=\sqrt{\frac{2}{\pi }}\frac{s}{\sigma }\exp \left[-\frac{z^2}{2{\sigma }^2}\right]\dot{z}& (4)\end{array}$

where, parameters m, c, k are, respectively, the mass, damping, and stiffness coefficients, and parameters ẍ, ẋ, x are quantities that describe the organization acceleration, velocity and displacement, and R is the restoring force and F(t) is the ambience excitation. Parameter α is the weighting value cogent the ratio of post-rubberband to initial stiffness. Parameters A, β, and γ are basic hysteresis shape control parameters. Parameter z is the hysteretic deportation, and n is the degree of the sharpness of yield. Strength and stiffness degradation coefficients are, respectively denoted by ν(δ_ν) and η(δ_η). Parameters 10 _i and x _two are Bouc-Wen hysteretic system displacement and the boosted displacement that considers slip-lock behavior. Parameter ε is the measure of the combined outcome of duration and severity of the free energy dissipated through hysteresis, σ is a measure out of the sharpness of the peak of the hysteresis, and δ _s measures the sideslip magnitude. All 10 parameters are essential to produce the mutual features of hysteretic beliefs. It would be very helpful if a small number of unspecified parameters for organization identification can exist reduced in that large numbers of parameters increase the dubiety of convergence for updating parameters in search space. It was proved that the redundancy of specific hysteresis parameters can be eliminated through mathematical transformations in the parameter space devised to freeze them without affecting the organisation response (Ma et al., 2004; Charalampakis and Koumousis, 2008a,b; Charalampakis and Dimou, 2010).

Structural System Modeling

Restoring forcefulness curves of reinforced or steel structures show complex hysteresis characteristics, revealing fabric non-linearity, cleft opening and endmost, bail and slip between steel bars and concrete and low cycle fatigue that consequence in structural strength and stiffness degradation. The hysteresis model used herein considers the specific case appropriate for both force and stiffness degradations, and sideslip-lock beliefs of restoring hysteretic forces of the structure. All the restoring forces of the structure are assumed to follow the BWBN hysteresis model. The structural system considered in this study is a shear type model subjected to ambience sinusoidal wave and ElCentro seismic excitation as input signals ẍ _g . The basis excitation move ẍ _thousand makes an integral transformation to be incorporated into the structural equations. The structural system mainly includes three lumped mass coupled subsystems, as shown in Figure 2. These masses are lumped at floor (floor g _i ), levels and these floors are assumed and constrained to only move laterally. The restoring strength R _i , in conjunction with the stiffness betwixt the side by side floors, are represented by dampers and springs, with the corresponding coefficients c _i and k _i , respectively. The time varying system yields energy dissipation due to the hysteretic behavior of the inner structure.

Figure 2. Structural lumped mass model.

For this iii story shear structure, the equations of motion are represented by Equations (8–10):

$\begin{array}{cc}{\mathrm{thou}}_3{\ddot{x}}_3+c_{\mathrm{three}}({\dot{x}}_3-{\dot{x}}_{\mathrm{two}})+R_3=0& (8)\end{array}$

$\begin{array}{cc}{m}_2{\ddot{x}}_2+c_{\mathrm{two}}({\dot{\mathrm{ten}}}_2-{\dot{x}}_{\mathrm{one}})+R_{\mathrm{two}}-c_3({\dot{x}}_3-{\dot{\mathrm{ten}}}_{\mathrm{ii}})-R_3=0& (9)\end{array}$

$\begin{array}{cc}{\mathrm{grand}}_{\mathrm{i}}{\ddot{x}}_{\mathrm{i}}+c_1({\dot{x}}_1-{\dot{\mathrm{ten}}}_g)+R_1-c_{\mathrm{two}}({\dot{\mathrm{10}}}_2-{\dot{\mathrm{ten}}}_1)-R_{\mathrm{ii}}=0& (10)\end{array}$

System Identification Theory

A typical system identification framework mainly has two components including system itself and organisation identification model. By defining an equivalence criterion, the parameters of the system identification model are updated via a comparison with the original system, until the organization identification model volition be eventually equivalent to the original arrangement. System identification approaches used in this newspaper include intelligent parameter varying based arroyo, GA based approach and transitional markov chain monte carlo (TMCMC) based approach. IPV arroyo employs ANNs and establishes a "Gray Box" system, where the original organization parameters are replaced by the parameters of ANNs, and the system structure is replaced past ANNs. For GA and TMCMC methods, they are used to optimize and identify the parameters of organisation model, which is regarded as an guess model of the original existent system. The theoretical background for the three approaches are presented in the following subsections.

Intelligent Parameter Varying Based Organisation Identification

Artificial neural network (ANN) is a non-linear and adaptive information processing system composed of large numbers of neuron units. A radial basis function (RBF) neural network is employed to build the intelligent parameter varying model. In ANN architecture, the Euclidean altitude between the clustering centre and input vector is calculated and the issue is activated to pass through the output layer. The activation function is Gaussian output office and is formulated as:

$\begin{array}{cc}{g}_j=\exp \left(-\frac{{\displaystyle {\sum }_{i=1}^n{(x_i-c_j)}^2}}{{\sigma }_j^{\mathrm{ii}}}\right)& (\mathrm{eleven})\end{array}$

where k _j is the output of the j ^th unit in the hidden layer, x _i is the input data fed to the network, c _j is the center of the j ^th unit in the input space, and σ _j is the width of the j ^th function. j = 1, two, ⋯ , m. Parameter m is the number of the centers of neurons, and due north represents the dimension of the input space. Linearly weighted summation of hidden layer node outputs produces the output nodes. Therefore, the output of the network is calculated past:

where w _j is the weight of the j ^thursday node.

The objective is to find a series of weights that minimize the foursquare of error betwixt the actual and desired network outputs, i.east.,:

$\begin{array}{cc}E(k)=\frac{1}{\mathrm{ii}}{\displaystyle \sum _{j=1}^{\mathrm{Due\; north}}{(d(\mathrm{one\; thousand})-y(\mathrm{yard}))}^2}& (13)\end{array}$

where d(k) is the desired output and y(1000) the bodily output of RBF and j = ane, ii, ⋯ , North, and N is the number of data sets. In that location are unremarkably 2 approaches for utilizing ANNs, i.e., supervised and unsupervised learning. In supervised approach the input and the referred output are usually known. The network and then processes the input values and makes a comparison between its resulting output and the desired output. The network system makes the errors propagate back through dissimilar layers, causing the arrangement to accommodate tuned weights which control the network. This process repeats over and over until all the weights are continually tweaked in an appropriate style. During the training process of a network the connexion weights are continually refined to a specific generalization level and a good network performance level. In unsupervised training, the organisation itself must then make up one's mind what features and how many features need to be extracted to group the input data, which is oftentimes referred to as cocky-organization or adaption. Herein, we use the non-supervised learning algorithm to acquire the centers and variance of radial basis office, and meanwhile the least mean squared fault is acquired by using supervised learning algorithm. The response of the hysteretic system is used every bit the desired signal, and the error between the desired signal and the imitation signal is back propagated to alter the weights and threshold values of neural network model.

Parametric system identification approaches accept been widely used but in well-nigh published literature in this area a priori knowledge of the characteristics depicting the behavior of restoring force is required. Not-parametric approaches generally do non need data beforehand but they typically lack directly associations between organization dynamics and associated model. When ANNs are implemented using the "Blackness Box" approach, niggling of the organization information might be obtained from the traditional techniques due to the fact that the "Black Box" just considers organisation input and output. Intelligent Parameter Varying (IPV) method preserves the benefits of both traditional parametric and non-parametric approaches, and utilizes the embedded radial basis function as the activation of neurons to estimate the constitutive characteristics of inelastic and hysteretic restoring forces for a multi degree of freedom structural system.

IPV technique, i.e., a grayness box approach (Figure 3) that incorporates the advantages of both "White Box" and "Black Box" approaches, was adult in such a mode that the model structure can be determined using the first principle (Equations 18–20), while not-linear and adaptive learning capabilities of ANNs can be used to identify the non-linear, fourth dimension varying system'due south dynamics (Equations 24–26) that would exist difficult to model and identify using the traditional "White Box" and "Black Box" (Saadat et al., 2003, 2004a,b, 2007).

Figure iii. System identification using greyness box.

A non-linear system with full state measurement represented past the Linear Parameter Varying (LPV) model structure is given by:

$\begin{array}{cc}\dot{x}=f_1(\mathrm{10},u)\cdot x+f_{\mathrm{ii}}(x,u)\cdot u& (14)\end{array}$

The IPV approach introduced herein would preserve the model structure inherent in Equation (14) without requiring a priori representations of non-linearities f _i(ten, u) and f ₂(ten, u). Instead, these terms would be represented past separate artificial neural networks grand _i(ten, u, west ₁) and g ₂(x, u, w ₂) every bit depicted in Equations (16, 17):

$\begin{array}{cc}\dot{x}={\mathrm{one\; thousand}}_1(\mathrm{10},u,w_1)\cdot x+{\mathrm{chiliad}}_2(\mathrm{ten},u,{\mathrm{west}}_2)\cdot u& (\mathrm{xvi})\end{array}$

Past modeling the non-linearities f ₁(10, u) and f ₂(x, u) via separate artificial neural networks grand ₁(ten, u, w ₁) and g _two(x, u, west _ii), the model structure (Equations sixteen, 17) is preserved. Therefore, the human relationship between the model construction and artificial neural network parameters is preserved. The structural model is preserved by incorporating ANN, preserving a portion of information of the structural model. The IPV approach preserves the straight clan between the structure of ANN and the organization dynamics, used for structural health monitoring for system identification.

Based on first principle, system dynamics can be transformed to the post-obit form:

$\begin{array}{cc}-c_{\mathrm{three}}({\dot{\mathrm{10}}}_3-{\dot{x}}_2)-R_3-g_3{\ddot{x}}_g=m_{\mathrm{three}}{\ddot{u}}_{\mathrm{iii}}& (\mathrm{xviii})\end{array}$

$\begin{array}{cc}-c_{\mathrm{ii}}({\dot{x}}_{\mathrm{ii}}-{\dot{x}}_{\mathrm{ane}})-R_2+c_{\mathrm{three}}({\dot{x}}_3-{\dot{x}}_2)+R_3-{\mathrm{one\; thousand}}_2{\ddot{\mathrm{10}}}_g={\mathrm{thousand}}_2{\ddot{u}}_2& (19)\end{array}$

$\begin{array}{cc}-c_1({\dot{x}}_1-{\dot{x}}_g)-R_1+c_2({\dot{x}}_2-{\dot{x}}_1)+R_{\mathrm{two}}-m_1{\ddot{x}}_{\mathrm{1000}}=m_1{\ddot{u}}_1& (20)\end{array}$

where u _one, u ₂, u ₃, respectively represent the relative displacements of each floor, i.e., u ₁ = x ₁−x _thou , u ₂ = 10 ₂−ten ₁₀₀₀ , u _iii = x _iii−x _g .

$\begin{array}{cc}-R_{\mathrm{ii}}+R_{\mathrm{three}}-m_{\mathrm{two}}{\ddot{x}}_{\mathrm{one\; thousand}}=m_{\mathrm{ii}}{ü}_2& (22)\end{array}$

$\begin{array}{cc}-R_{\mathrm{i}}+R_2-m_1{\ddot{\mathrm{10}}}_g=m_{\mathrm{ane}}{ü}_{\mathrm{one}}& (23)\end{array}$

The stiffness and damping terms are lumped into restoring forces, since in the hysteresis models the restoring force R associates with the lateral relative displacement ten _r and the restoring deportation z, where z is expressed by the function of lateral relative velocity ẋ _r .

The modeling for the restoring forces using radial footing function based neural network is as follows:

$\begin{array}{cc}{\widehat{R}}_{\mathrm{ii}}={\mathrm{thousand}}_2({ü}_{\mathrm{ii}},{\ddot{x}}_{\mathrm{yard}},{\widehat{R}}_3)& (25)\end{array}$

$\begin{array}{cc}{\widehat{R}}_1={\mathrm{grand}}_1({ü}_1,{\ddot{x}}_{\mathrm{grand}},{\widehat{R}}_{\mathrm{two}})& (26)\end{array}$

where, ${\widehat{R}}_1,{\widehat{R}}_{\mathrm{two}},{\widehat{R}}_3$ represent, respectively, the identified restoring forces through the training of ANN as shown in Figure 4.

Effigy 4. IPV for structural system modeling.

Genetic Algorithm Based System Identification

Genetic algorithm (GA) is an approach that searches the global optimal solution past simulating a natural development process. The objective role can be formulated equally the normalized mean square error (MSE) of the predicted time history ỹ(t|p) every bit compared to the reference time history y(t). Herein, the acceleration response signal is employed as the fourth dimension history series. The purpose of the post-obit optimization arroyo is to minimize the difference (or the error) betwixt predicted time history and the referred time history. The hysteretic structural arrangement objective office is introduced below:

$\begin{array}{cc}OF(p)=\frac{{\displaystyle {\sum }_{i=\mathrm{one}}^n{\left(y(t_i)-\tilde{y}\left(t_i|p\right)\right)}^{\mathrm{ii}}}}{\mathrm{Due\; north}{\sigma }_y^2}& (27)\end{array}$

where p is a parameter vector, ${\sigma }_y^2$ the variance of the reference history, and N the number of points used. Sum of iii dispatch response bespeak differences of the hysteretic structural organization is used as the objective role. The optimization problem tin can exist stated as the minimization of the objective office OF(p) when the parameter vector has the post-obit side constraints as:

where x _LB and x _UB are vectors defining the lower and the upper values of the model parameters, respectively. The basic strategy for the parameter identification using GA is shown in Figure 5. GA operates starting from a population of the potential solutions to a representive problem, and one population is composed of numbers of individuals coded by genes. Each individual is chromosomes with the characteristics of entity. GA initializes on a population of individuals (coded candidate solutions to the problem) that are manipulated by some operators such as selection, crossover, and mutation. In brusque, the pick procedure drives the search direction toward the region of best individuals, and the cantankerous operator combines individuals to generate offsprings.

Figure v. A flow chart of system identification using GA.

If information technology is indispensable to make choice and crossover operators converge toward the optimum, mutation alters one or more than factor values (individuals) in a chromosome from its initial state, thus, maintaining genetic diversity from ane generation of a population to the adjacent. In this manner, a complete exploration of global search space is forced by algorithm within the search space. Each individual in the population is represented as chromosome, indicating the drove of parameters are supposed to identify. GA adopts an elitist strategy, which consists of the preservation of the most fit individuals obtained in the current generation. Population representation and initialization generate population and individuals, and the initialized value will be assigned to the parameter space for solving the hysteretic organisation model. The fitness function is established by using the prediction error between the imitation indicate response and predicted signal response. After a series of successive mathematical operators for optimization, the next generational loop begins.

Transitional Markov Concatenation Monte Carlo Based System Identification

Markov Chain Monte Carlo (MCMC) is an analytic approach which replaces numerical integration through summation over numbers of samples generated from iteration. A Markov chain is a stochastic procedure where i state is transformed to some other country afterwards a sufficiently long sequence of transition procedure. The next state is conditionally based on the last state. A central property of Markov chain is that the starting country has no influence on the state of the chain via a series of sequential transitions. The concatenation reaches its steady state at a specific signal where it reflects sampling distribution from stationary status. The principle of Monte Carlo simulation is applied for the integration to gauge the expected complex distribution status of numbers of samples. By increasing the number of samples, the approximation accurateness can be measured and achieve a desired value, which mainly depends on the independence of the samples.

Markov chain involves a stochastic sequential process where a series of states tin exist sampled from some stationary distributions while Monte Carlo sampling can make an estimation of various characteristics of a specific distribution. The goal of MCMC is to design a Markov chain to meet the target distribution of the chain which is what we are interested in sampling from.

Transitional Markov Chain Monte Carlo Theory (TMCMC) is introduced to avoid the difficulty of sampling from complicated target probability distributions (e.one thousand., multimodal PDFs, PDFs with apartment manifold, and very peaked PDFs) but sampling from a series of intermediate PDFs that converge to the target PDF and are easier to sample.

Bayesian Inference describes a procedure of solving posterior density functions given the likelihood and prior probability. The target probabilistic model tin can be depicted by M, D is the data acquired from the system, and the uncertain parameters of the model are described equally θ. Sampling from the posterior PDF of θ conditioned on D is the aim of the Baysian model updating, which is given as:

$\begin{array}{l}\begin{array}{ccc}f\left(\theta |M,D\right)& =& \frac{f\left(D|\mathrm{Thou},\theta \right)\cdot f\left(\theta |\mathrm{1000}\right)}{f\left(D|K\right)}\end{array}\\ \begin{array}{cc}=& \frac{f\left(D|M,\theta \right)\cdot f\left(\theta |K\right)}{{\int }^{\text{}}f\left(D|M,\theta \right)\cdot f\left(\theta |M\right)\cdot d\theta }\end{array}& (29)\end{array}$

where f(θ|M) is the prior PDF of θ, f(D|M, θ) is the likelihood of D given θ, and f(D|Yard) is the evidence of the model M.

Bayesian model updating generally employs simulation based methods in that information technology is effective to obtain samples from f(θ|M, θ), which tin can be estimated at a specific quantity of interest E(g|1000, D) based on the Law of Large Number.

$\begin{array}{cc}\mathrm{Eastward}\left(k|M,D\right)\approx \frac{1}{\mathrm{Northward}}{\displaystyle \sum _{k=\mathrm{ane}}^{\mathrm{North}}\mathrm{thousand}\left({\theta }_m\right)}& (30)\end{array}$

where {θ _yard :k = 1, two, ⋯ , N} represents a set up of Northward samples from f(θ|Thou, D). Consider the equation every bit follow:

$\begin{array}{cc}f(\theta |M,D)\propto f(\theta |\mathrm{Grand})·f(D|\mathrm{1000},\theta )& (31)\end{array}$

Information technology is usually difficult to sample from f(θ|M, D) using Importance Sampling (IS) and Metropolis–Hastings (MH) in that it is non so easy to sympathise the geometry of the likelihood f(D|Yard, θ). To converge to the target PDF f(θ|Grand, D) from the prior PDF f(θ|K), a series of intermediate PDFs are constructed as the following:

$\begin{array}{l}\begin{array}{c}{f}_j(\theta )\propto f(\theta |G)\cdot f{(D|M,\theta )}^{p_j},j=0,1,\cdots ,k\end{array}\\ \begin{array}{c}0=p_0<p_1<\cdots <p_m=\mathrm{i}\end{array}& (32)\end{array}$

Note that f ₀(θ) = f(θ|Grand), f _thou (θ) = f(θ|Yard, D).

where the index j denotes the phase number. Although the geometry irresolute from f(θ|G) to f(θ|M, D) is large, the status of 2 irresolute adjacent intermediate PDFs is minor. Information technology is efficient to sample from f _j+1(θ) co-ordinate to the previous sample from f _j (θ) through this minor change.

The office f _j (θ) is used to extract samples and brand an estimation of the PDF itself as a kernel density part (KDF), a combination of weighted Gaussian functions centered at the samples. The kernel density function tin can be regarded every bit the proposal PDF of the MH method to sample from f _j+1(θ). This will subsequently and ultimately result in f(θ|M, D) samples. This arroyo is chosen adaptive City–Hastings (AMH) algorithm. Given that the proposal PDF (KDF) function is fixed, rendering the MH method is as similar as IS, not efficient in high dimension state of affairs.

It is a totally different strategy for TMCMC to larn f _j+1(θ) samples based on f _j (θ) samples, KDF method is replaced by a resampling algorithm. It covers a battery of resampling stages, with each stage completing the following, given Due north _j samples from f _j (θ), depicted by {θ _{j, k} :1000 = 1, ⋯ , N _j }, acquire samples from f _j+1(θ), depicted by{θ _{j+one, m} :chiliad = ane, ⋯ , N _j+1}. Information technology can be calculated by the following in a more easier way, with the samples {θ _{j, thousand} :grand = 1, ⋯ , N _j } from f _j (θ). The "plausibility weights (w(θ _{j, 1000} ))" of these samples regarding f _j+1(θ) can be computed by:

$\begin{array}{l}\begin{array}{ccc}w({\theta }_{j,\mathrm{thou}})=\frac{f({\theta }_{j,k}|M)f{(D|M,{\theta }_{j,k})}^{p_{j+\mathrm{one}}}}{f({\theta }_{j,k}|M)f{(D|\mathrm{Thousand},{\theta }_{j,k})}^{p_j}}& =& f{(D|\mathrm{Thousand},{\theta }_{j,k})}^{p_{j+1}-p_j},\end{array}\\ \begin{array}{ccc}k& =& 1,\cdots ,{\mathrm{Northward}}_j\end{array}& (33)\end{array}$

Based on the normalized weights, the uncertain parameters tin can exist resampled, i.e. let: θ _{j+1, 1000} = θ _{j, l} and $\mathrm{due\; west}.p.\frac{\mathrm{westward}({\theta }_{j,l})}{\sum _{l=\mathrm{ane}}^{N_j}w({\theta }_{j,l})}$ k = 1, ⋯ , N _j+ane

where "with probability" is represented by w.p. and pacifier index is denoted as l. It is shown that if N _j and N _j+1 accomplish a relatively big quantity, {θ _{j+1, k} :one thousand = 1, ⋯ , N _j+1} will be distributed as f _j+1(θ). Moreover, due west(θ _{j, thousand} ) is expected equally the following value:

$\begin{array}{l}E\left[w({\theta }_{j,\mathrm{thousand}})\right]={{\displaystyle \int }}^{\text{}}w(\theta )\cdot f_j(\theta )\cdot d\theta \\ ={{\displaystyle \int }}^{\text{}}f{\left(D|G,\theta \right)}^{p_{j+1}-p_j}\cdot f_j(\theta )\cdot d\theta \\ ={{\displaystyle \int }}^{\text{}}f{\left(D|M,\theta \right)}^{p_{j+1}-p_j}\cdot \frac{f(\theta |G)f{(D|M,\theta )}^{p_j}}{f(\theta |M)f{(D|M,\theta )}^{p_j}d\theta }\cdot d\theta \\ \begin{array}{cc}=& \frac{{\int }^{\text{}}f(\theta |\mathrm{Thou})f{(D|M,\theta )}^{p_{j+\mathrm{i}}}d\theta }{{\int }^{\text{}}f(\theta |\mathrm{Grand})f{(D|\mathrm{One\; thousand},\theta )}^{p_j}d\theta }\end{array}& (34)\end{array}$

Therefore, $\sum _{\mathrm{yard}=\mathrm{i}}^{N_j}w({\theta }_{j,k})/{\mathrm{North}}_j$ is the automatically unbiased estimation made for $\frac{\int {f(\theta |K)f(D|M,\theta )}^{p_{j+1}}d\theta }{\int {f(\theta |M)f(D|M,\theta )}^{p_j}d\theta }$. According to the results above, the following method is used to sample from f(θ|1000, D) and make an interpretation of f(D|M).

More than precisely, with probability $\mathrm{west}({\theta }_{j,k})/\sum _{l=1}^{{\mathrm{Northward}}_j}w({\theta }_{j,l})$, by using a covariance matrix equal to the scaled version of the estimated covariance matrix of f _j+1(θ), a Markov chain sample in the yard ^th concatenation tin can be generated from a Gaussian proposal PDF centered at the current sample of the k ^thursday chain:

$\begin{array}{l}{\displaystyle \sum _j^{}={\beta }^{\mathrm{two}}{\displaystyle \sum _{\mathrm{1000}=1}^{{\mathrm{North}}_j}w\left({\theta }_{j,\mathrm{one\; thousand}}\right)\left\{{\theta }_{j,k}-\left[\frac{{\displaystyle {\sum }_{j=1}^{N_j}w\left({\theta }_{j,\mathrm{fifty}}\right){\theta }_{j,\mathrm{50}}}}{{\displaystyle {\sum }_{j=1}^{{\mathrm{Due\; north}}_j}w\left({\theta }_{j,l}\right)}}\right]\right\}}}\\ \times {\left\{{\theta }_{j,\mathrm{chiliad}}-\left[\frac{{\displaystyle {\sum }_{j=1}^{{\mathrm{Northward}}_j}w\left({\theta }_{j,\mathrm{fifty}}\right){\theta }_{j,\mathrm{50}}}}{{\displaystyle {\sum }_{j=1}^{N_j}w\left({\theta }_{j,l}\right)}}\right]\right\}}^T& (35)\end{array}$

where β is the prescribed scaling factor, $\sum _j=productof{\beta }^2$ and the estimated covariance of f _j+1(θ). The rejection rate is called equally the value β, and MCMC may probably achieve a larger value appropriately. The value 0.2 is institute to exist more reasonable for the scaling parameter β.

Information technology is essential to choose {p _j :j = 1, ⋯ , m−1}. The larger value of p is desirable to make the transition between the intermediate adjacent PDFs smoother. The number of intermediate stages achieves a huge value if the increase of p-values has dull change rates.

The degree of uniformity of the plausibility weights {w(θ _{j, k} ):chiliad = 1, ⋯ , Due north _j } can appropriately betoken how close f _j+1(θ) approaches f _j (θ), so p _j+1 should be chosen so that the coefficient of variation (COV) of the plausibility weights can exist equivalent to a prescribed threshold. The Bayesian inference framework for system identification is established (Figure 6) for structural model updating, which is regarded as a determinant reason for choosing the well-nigh suitable model parameters related to the hysteretic beliefs of the structures by minimizing the deviation between the predicted structural response and the simulated structural response. For the hysteretic structural model, the uncertain model parameters are selected as the ones that need to be updated through Bayesian inference (Equation 31) by drawing samples of parameters from the posterior PDF of parameters.

Figure 6. TMCMC based system identification strategy.

The TMCMC based Bayesian Updating algorithm is coded and implemented to constitute the computational surround via exchange of data between the model and the algorithm. The development of parameter updating process is that, the samples from the prior PDFs are approximately uniformly distributed in the model parameter space at the first stage (p ₀ = 0). Through applying Bayesian inference with TMCMC probabilistic simulations, the samples somewhen populate well in the high probability region of the posterior PDFs close to the true model parameters at the terminal stage (p _chiliad = 1).

TMCMC approach is employed to make an identification of the parameters of Bouc-Wen grade models, henceforth represented past the vector θ ≤ Θ⊆R ^d . The advisable pick of θ reflects the corresponding not-linear and hysteretic behavior of construction. By applying Bayesian model updating, the major advantage is that the result gives a probability distribution expressing the likelihood probability distribution of different parameters rather than yielding a unmarried value for θ. It is clear that the evolution of the model parameter variation represents the Bayesian inference process.

To make a brake of the parameter space Θ, two side constraint vectors θ_min and θ_max are defined such that:

$\begin{array}{cc}{\theta }_{\min }(i)\le \theta (i)\le {\theta }_{\max }(i)\mathrm{one}\le i\le d& (36)\end{array}$

The vector θ has specific constraints such that the generated initial samples can decide the feasible values that the parameters can take. Past defining prior PDF of θ a uniform distribution between the likelihood PDF and side constraints are regarded as the prediction error assumed every bit Gaussian role distributed with unknown variance and null hateful. The prediction fault is described as the mistake between the predicted system response and the simulated organization response given by:

$\begin{array}{cc}f(D|\theta )={\displaystyle \prod _{i=\mathrm{ane}}^l\frac{1}{{\sigma }_{acc}\sqrt{2\pi }}\exp \left[-\frac{1}{2{\sigma }_{acc}^2}{\left(\frac{\mathrm{ten}(t_i)-\widehat{x}(t_i|\theta )}{{\mathrm{Southward}}_{acc}(t_i)}\right)}^2\right]}& (37)\end{array}$

where ${\sigma }_{acc}^{\mathrm{two}}$ is the variance of the prediction errors and Due south _acc is the weighting function used to normalize the acceleration response of the hysteretic arrangement. To achieve computational convenience, the log-likelihood function lnf(D|θ) is employed in the actual implementation of the TMCMC algorithm every bit:

$\begin{array}{l}\begin{array}{ccc}\mathrm{50}\mathrm{due\; north}f(D|\theta )=& -& \frac{\mathrm{one}}{2}{N}_t\ln (2\pi )-N_t\ln ({\sigma }_{acc})\end{array}\hfill \\ \begin{array}{cc}-& \frac{1}{2{\sigma }_{acc}^2}{\left(\frac{x(t_i)-\widehat{x}(t_i|\theta )}{{\mathrm{Due\; south}}_{acc}(t_i)}\right)}^2\end{array}\hfill & (38)\end{array}$

The j ^thursday stage of parameter development process by correspondingly choosing the values of p _j can exist shown equally the contours of PDF f _j (θ).

Numerical Analysis of Arrangement Identification

Parameter Settings

Tabular array ane lists the parameter assignments of the structural system associated with BWBN hysteresis model for different cases. The mass coefficient of each flooring is ane kg, stiffness coefficient 10 N/yard (sin) and 20 Due north/m (ElCentro), damping coefficient 0.05 Northward/(1000/s) for each floor, respectively.

Tabular array 1. Parameter assignment for different cases.

For hysteresis model, A = 1, α = 0.01, β = two, γ = ii, due north = one (IPV), n = 1.5 (GA and TMCMC), δ_ν = 0, δ_ν1 = one.v (sin) and 3 (ElCentro), δ_η = 0, δ_η1 = 1.five (sin) and 3 (ElCentro), σ = 0.01, δ _s = 0.05. Damage occurrences are causeless at xxx south, 60 south, 100 south (sin) and 10 s, twenty due south, 40 s (ElCentro) when both the strength degradation factor δ_υ and stiffness degradation factor δ_η modify to δ_υ1and δ_η1, respectively. For GA and TMCMC approaches, parameter initial values are assigned betwixt the corresponding lower and upper premises before the optimization processes. Parameters are updated between the 2 bounds of parametric searching space, and eventurally achieve the exact values.

Numerical Analysis Results

Figures vii, 8 bear witness the restoring force identification results when the construction is subjected to sinusoidal bespeak and ElCentro signal, respectively. Black curve represents the restoring force of the original structural system while the red bend represents the identified restoring forcefulness using IPV, GA, and TMCMC approaches. For the case of sinusoidal excitation, the identification results show that the degradation phenomenon of restoring forces of 2nd−3rd floor and 1st−2nd flooring are more axiomatic compared with footing-1st floor past using all of the iii approaches. For the example of ElCentro excitation, the identification results show that the degradation phenomenon of restoring forces existing in all adjacent floors by using IPV and GA methods is more axiomatic compared with that using TMCMC method. The hysteresis curves rotate appropriately when damage occurs. More results in details are discussed in the next section.

Figure 7. System identification results (Sin) (A–C) IPV, (D–F) GA, (G–I) TMCMC.

Figure 8. Arrangement identification results (ElCentro) (A–C) IPV, (D–F) GA, (1000–I) TMCMC.

Discussions

Choice of Betoken Excitation Type and Objective Role

Both sinusoidal excitation signal and ElCentro excitation signal are used as input signals to the organization identification of hyestetic system. The frequency of sin bespeak is unmarried and history curve is smooth and periodic variation. The ElCentro wave is stochastic, non-periodic, and non-steady state, and this represents the other type of excitation which is totally counter to harmonic moving ridge such as sin betoken. These ii types of signals are employed to the response assay and verification of the generalization capability of three different algorithms for identifying restoring forces of hysteretic structure. In addition, the establishment of objective functions employed in the GA and TMCMC are formulated through the combination of desired and false dispatch signal differences of all three degrees of freedom.

Parameter Pick for Organization Identification

Table 2 shows the parameter identification mistake using GA and TMCMC for the example of noise free, SNR = 30 and SNR = 10 (SNR = signal dissonance ratio). These error represents the relative error between the "true value" and "identified value." All errors are no more than 10%. By and large for the cases with higher SNR they perform better parameter identification results since the disturbance of noise to signal is low. From Figures vii, eight, and Table 2, for the noise gratuitous case, it is shown that during the degradation procedure (harm evolution), force deposition gene δ_υ changes from 0 to 1.4695 (sin,GA) /3.0130 (ElCentro,GA) /ane.5913 (sin,TMCMC) /2.9890 (ElCentro,TMCMC), and stiffness degradation factor δ_η changes from 0 to 1.5370 (sin,GA)/two.9142 (ElCentro,GA)/1.5400 (sin,TMCMC)/two.7357 (ElCentro,TMCMC).

Table 2. Parameter identification fault for different cases.

The hysteresis loop rotates clockwise with a sure degree, indicating large not-linearity and considerable deposition. However, from the identification results, it is also institute that in the sin excitation case, the hysteresis loop exhibits meliorate slip-lock phenomenon, and can absorb more energy than the case of ElCentro excitation. Racket corrupted cases take like parameter identification results.

Tabular array 3 shows the analysis of correlation coefficients for different cases. For the cases with small-scale SNR, the identification effectiveness is not relatively good compared with dissonance free and higher SNR (R-t is the restoring force-time relationship, and R-x is the restoring forcefulness-relative displacement relationship).

Tabular array 3. Correlation coefficient analysis for different cases.

In this paper, IPV is a radius basis neural network which is a completely data-driven approach, while GA and TMCMC are non fully information-driven in that they assign the identification model inside specific initial parameter intervals and and then update and search to optimize structural model.

IPV method performs better identification results due to its adaptive learning capability and anti noise belongings in the noise environment. From computational time, it tin can be concluded that the IPV approach is a more efficient identification method compared with GA and TMCMC methods. Concretely, for the case of noise free, the computational time for GA and TMCMC are 9.59 and 760.51% longer than IPV (sin), and 8.77 and 694.40% longer than IPV (ElCentro). Like results are too shown in racket corrupted cases. This demonstrates IPV method has higher computational efficiency than GA and TMCMC approaches.

Force and stiffness deposition parameters are both very important in the hysteresis model, which determine the hysteretic behavior of structural systems. Table four lists v groups of parameter assignments regarding the variance of strength and stiffness degradation coefficients. The objective of setting these cases is studying the influence of alter of degradation parameters on system identification accuracy.

Tabular array 4. Degradation parameter assignment for different cases.

To simulate damage occurrence, for IPV approach, strength deposition factor δ_υ and stiffness deposition factor δ_η change from one.v to 1.8, ii.i, 2.four, and two.7, respectively (sin), while change from three to three.3, 3.vi, 3.ix, and 4.two, respectively (ElCentro). For GA (sin) method, for the starting time 3 groups, the true value of force deposition gene δ_υ and stiffness degradation factor δ_η proceed stock-still value one.5 merely the lower and upper bound change from 0.1–3 to 0.5–two.5 and one–2, respectively. For the terminal ii groups, the truthful value of strength deposition factor δ_υ and stiffness degradation gene δ_η are ane.8 and 2.i, respectively but the lower and upper leap correspondingly change from 0.one to 3.6 and 0.one to four.2, respectively.

For GA (ElCentro) method, for the first three groups, the true value of strength degradation factor δ_υ and stiffness degradation factor δ_η go on fixed value 3 merely the lower and upper bound alter from 0.one–half-dozen to 1–5 and 2–4, respectively. For the last ii groups, the true value of strength degradation factor δ_υ and stiffness degradation factor δ_η are iii.three and iii.6, respectively only the lower and upper bound correspondingly change from 0.1 to 6.6 and 0.one to 7.2, respectively. TMCMC parameter assignment has similar cases.

Table 5 shows the influence of change of forcefulness and stiffness degradation parameter on system identification accuracy indicated by correlation coefficients using IPV method. For sin excitation, as the strength and stiffness deposition parameter increase from 1.5 to two.seven, the correlation coefficients do non increase or decrease for the aforementioned case, and the computational time also keep almost unchanged: (sin) Group1: 85.eleven–95.67 s, Group ii: 89.01–104.78 s, Group iii: 88.97–107.37 s, Grouping 4: 83.12–100.20 s, Grouping v: 86.09–104.55 due south. (ElCentro) Group 1: 93.33–108.98 due south, Group 2: 92.17–115.90 s, Group three: xc.00–120.03 due south, Group four: 87.21–107.69 s, Group 5: 89.11–111.eleven s.

Table v. Correlation coefficient analysis (IPV).

Tabular array half dozen shows the influence of change of strength and stiffness deposition parameter on system identification accuracy indicated by correlation coefficients using GA method. For sin excitation, when the strength and stiffness deposition parameter value is stock-still at i.5, as premises (interval) modify from 0–3 to i–2, the correlation coefficients approach 1, indicating the organisation identification accurateness improves with the bound interval decreasing. The closer the bounds approach to the true value, the more authentic and deterministic the organization identifiion results are. Accordingly, the computational time decreases with the bound interval decreasing. For details: Group ane: 901.31–927.76 s, Group 2: 895.66–922.22 s, Grouping 3: 889.73–914.73 s. When the true value of forcefulness and stiffness degradation increases from 1.5 to two.ane with given bounds, the identification accuracy decreases, and the computational time increases accordingly. Like cases tin also be establish for ElCentro excitation betoken.

Table half dozen. Correlation coefficient analysis (GA).

Table 7 shows the influence of change of strength and stiffness degradation parameter on system identification accurateness indicated by correlation coefficients using TMCMC method. For sin excitation, when the strength and stiffness degradation parameter value is stock-still at 1.5, as bounds (interval) change from 0–3 to 1–2, the correlation coefficients approach 1, indicating the arrangement identification accuracy improves with the bound interval decreasing.

Tabular array 7. Correlation coefficient analysis (TMCMC).

The closer the premises approach to the truthful value, the more than accurate and deterministic the system identifiion results are. Accordingly, the computational time decreases with the bound interval value decreasing, equally shown in detailed cases: Group 1: 64812.36–66409.77 s, Group ii: 63508.90–65546.11 s, Group three: 62109.67–64111.67 s. When the true value of strength and stiffness degradation increases from one.5 to 2.1 with given bounds, the identification accurateness decreases, and the computational fourth dimension increases accordingly. Like cases can also be plant for ElCentro excitation indicate.

Comparing of Three Different Algorithm Principles

Intelligent Parameter Varying approach uses radial basis role to map the complex input signal to loftier dimensional signal, and it is a data driven machinery. By using appropriate error back propagation machinery, this method tin can blueprint a good neural network compages to process considerable corporeality of data or loftier parameter dimension, especially for system identification application. Genetic algorithm and Transitional Markov Chain Monte Carlo approaches are not-data-driven intelligent optimization algorithms. Genetic algorithm optimizes the parameters by using selection, crossover, and mutation operators through elitist strategy. Transitional Markov chain Monte Carlo method employs model updating to optimize parameters through applying Bayesian inference with TMCMC probabilistic simulations, and the samples eventually populate well in the loftier probability region of the posterior PDFs close to the true model parameters through a series of intermediate updating processes. The latter 2 methods are related to optimization theory, and may non perform well (tapped in local optimum) especially when the parameter dimension is relatively big. Therefore, it is very significant to conduct a comparative study on arrangement identification of hysteretic structures using Intelligent Parameter Varying, Genetic Algorithm and Transitional Markov Chain Monte Carlo methods.

The higher up discussions regarding the choice of signal excitation blazon, parameter choice for system identification and comparison between three unlike principles for arrangement identification has demonstrated the necessity, feasibility and importance of this inquiry. It besides illustrates the generalization of this research and proves the superiority using IPV over GA and TMCMC for organisation identification of hysteretic structures.

Conclusions

To improve sympathise the hysterically degrading, structural systems using an earlier version of smoothly varying Bouc-Wen-Baber-Noori hysteresis model in this research, a detailed clarification of BWBN hysteresis model is presented, and the restoring force and the associated system variables are analyzed using non-linear differential equations containing different parameters. By choosing the parameters in a suitable mode, information technology is possible to generate a big variety of different shapes of the hysteresis loops. A three floor shear construction is modeled which composes of three adjacent subsystems by associating organization kinetic equations, restoring force expression and BWBN hysteresis model. By using BWB-Noori model, a new scheme is proposed to finer and efficiently rails and approximate the hysteretic restoring forces using intelligent parameter varying approach, genetic optimization algorithm and the transitional Markoff chain Monte Carlo simulation. Most importantly, comparative study by using these approaches for different cases is demonstrated through parameter fault analysis and correlation coefficient analysis of organization identification of fourth dimension varying degrading structures. Major findings are summarized in the following statements.

(one) BWBN hysteresis model is a smoothly varying differential mathematical model, and it can reflect highly non-linear and gradual hysteretic degradation with slip lock behavior observed in numerous structural and mechanical systems, and this model can exist widely used to predict the response of degradation phenomena of general structures.

(2) When employing organization identification and parameter/model updating approach, the initial parameter spaces of the hysteretic system should be well assigned to satisfy the requirement of reliable arrangement identification process. Tracking of the restoring forces for the hysteretic system using system identification approaches can accurately estimate the changing of restoring forcefulness condition in time, i.east., the rotation of hysteresis loops indicates structural degradation due to abrupt damages. This proves that organization identification techniques tin be used as powerful tools for detecting the harm/deposition for structural wellness monitoring applications. Stiffness and damping terms are lumped into restoring forces represented by structural non-linearity, and this constructs effective IPV modeling.

(3) IPV, GA, and TMCMC methods are employed for the system identification of BWBN model, and a comparative study is conducted for the verification of the effectiveness of these approaches. The results evidence higher SNR cases have better correlation and smaller parameter errors. From the correlation analysis, we also know that IPV has improve anti-noise capability than GA and TMCMC.

(four) Qualitative comparison regarding the computational time of these three different algorithms are ranked for different cases, i.due east., IPV Based system identification approach < GA based system identification method < < TMCMC based arrangement identification approach. This demonstrates that compared with traditional parameter optimization and statistical methods, IPV approach is a promising, efficient and effective way for system identification and Structural Health Monitoring applications.

(5) IPV technique using the RBF based ANNs has its superior advantages over the GA based identification and the TMCMC based identification techniques for its fully data-driven adaptive learning power for loftier dimensional data. Proper blueprint of parameter initial premises can improve the computational efficiency for GA and TMCMC approaches. The GA based identification may accept relatively uncertain values for the randomness of genetic operations (selection, crossover, and mutation), while the TMCMC algorithm is based on the sampling technique that is non as effective and is uncertain, especially for the case that the system has a relatively big number of parameters.

Author Contributions

YZ as Ph.D. educatee. MN as main supervisor. WA supervisor. TA was invited, due to his expertise in the expanse of Genetic Algorithms, to join the newspaper as a co-writer. TA carefully reviewed the technical analysis and independently carried out a system ID using Genetic Algorithm. His contributions were valuable in this regard and thus, his proper name is added in the newly revised version.

Conflict of Interest Statement

The authors declare that the research was conducted in the absence of whatsoever commercial or fiscal relationships that could be construed equally a potential conflict of involvement.

References

Abouelwafa, M. Due north., El-Gamal, H. A., Mohamed, Y. Due south., and Altabey, W. A. (2014). An expert organisation for life prediction of woven-roving GFRE closed end thick tube subjected to combined angle moments and internal hydrostatic pressure using artificial neural network. Int. J. Adv. Mater. Res. 845, 12–17. doi: x.4028/www.scientific.net/AMR.845.12

CrossRef Total Text | Google Scholar

Altabey, Westward. A. (2014). "Vibration analysis of laminated composite variable thickness plate using finite strip transition matrix technique and MATLAB verifications," in MATLAB- Particular for Engineer, ed K. Bennett (InTech), 583–620.

Altabey, W. A. (2016). FE and ANN model of ECS to simulate the pipelines suffer from internal corrosion. Struct. Monit. Mainten. 3, 297–314. doi: 10.12989/smm.2016.3.3.297

CrossRef Full Text

Altabey, W. A. (2017a). An exact solution for mechanical behavior of BFRP Nano-thin films embedded in NEMS. J. Adv. Nano Res. 5, 337–357. doi: 10.12989/anr.2017.v.iv.337

CrossRef Total Text

Altabey, W. A. (2017b). Free vibration of basalt fiber reinforced polymer (FRP) laminated variable thickness plates with intermediate elastic support using finite strip transition matrix (FSTM) method. J. Vibroeng. 19, 2873–2885. doi: 10.21595/jve.2017.18154

CrossRef Full Text | Google Scholar

Altabey, W. A. (2017c). Prediction of natural frequency of basalt cobweb reinforced polymer (FRP) laminated variable thickness plates with intermediate rubberband support using artificial neural networks (ANNs) method. J. Vibroeng. 19, 3668–3678. doi: x.21595/jve.2017.18209

CrossRef Full Text | Google Scholar

Altabey, W. A. (2017d). Delamination evaluation on basalt FRP blended pipe by electrical potential change. J. Adv. Aircraft Spacecraft Sci. four, 515–528. doi: 10.12989/aas.2017.four.five.515

CrossRef Full Text

Altabey, W. A. (2017e). EPC method for delamination cess of basalt FRP pipe: electrodes number effect. J. Struct. Monit. Mainten. 4, 69–84. doi: 10.12989/smm.2017.4.1.069

CrossRef Full Text

Altabey, W. A. (2018). Loftier functioning estimations of natural frequency of basalt FRP laminated plates with intermediate rubberband support using response surfaces method. J. Vibroeng. 20, 1099–1107. doi: 10.21595/jve.2017.18456

CrossRef Full Text | Google Scholar

Altabey, West. A., and Noori, M. (2017a). Detection of fatigue cleft in basalt FRP laminate composite pipe using electrical potential change method. J. Phys. Conf. Ser. 842:012079. doi: 10.1088/1742-6596/842/ane/012079

CrossRef Full Text | Google Scholar

Altabey, Due west. A., and Noori, M. (2017b). Fatigue life prediction for carbon fibre/epoxy laminate composites under spectrum loading using ii different neural network architectures. Int. J. Sustain. Mater. Struct. Syst. 3, 53–78. doi: 10.1504/IJSMSS.2017.092252

CrossRef Full Text | Google Scholar

Altabey, W. A., and Noori, Grand. (2018). Monitoring the water absorption in GFRE pipes via an electrical capacitance sensors. J. Adv. Aircraft Spacecraft Sci. 5, 411–434. doi: 10.12989/aas.2018.v.4.499

CrossRef Full Text | Google Scholar

Baber, T. T., and Noori, M. North. (1985). Random vibration of degrading, pinching systems. J. Eng. Mech. 111, 1010–1026. doi: ten.1061/(ASCE)0733-9399(1985)111:8(1010)

CrossRef Full Text | Google Scholar

Baber, T. T., and Noori, 1000. Due north. (1986). Modeling general hysteresis behavior and random vibration application. J. Vib. Acoust. 108, 411–420. doi: 10.1115/i.3269364

CrossRef Full Text | Google Scholar

Baber, T. T., and Wen, Y. K. (1980). Stochastic Equivalent Linearization for Hysteretic, Degrading, Multistory Strutctures. Academy of Illinois Engineering Experiment Station; College of Engineering science; Academy of Illinois at Urbana-Champaign.

Google Scholar

Beck, J. 50., and Au, S. K. (2002). Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation. J. Eng. Mech. 128, 380–391. doi: 10.1061/(ASCE)0733-9399(2002)128:four(380)

CrossRef Total Text | Google Scholar

Behmanesh, I., and Moaveni, B. (2014). Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite chemical element model updating. J. Struct. Control Hlth. Monit. 22, 463–483. doi: 10.1002/stc.1684

CrossRef Full Text | Google Scholar

Bouc, R. (1967). "Forced vibration of mechanical systems with hysteresis," in Proceedings of the Quaternary Conference on Non-linear Oscillation (Prague).

Brewick, P. T., Masri, S. F., Carboni, B., and Lacarbonara, W. (2016). Information-based nonlinear identification and constitutive modeling of hysteresis in NiTiNOL and steel strands. J. Eng. Mech. 142, ane–17. doi: 10.1061/(ASCE)EM.1943-7889.0001170

CrossRef Full Text | Google Scholar

Chan, R., Yuen, J., Lee, E., and Arashpour, 1000. (2015). Application of nonlinear-autoregressive-exogenous model to predict the hysteretic behaviour of passive control systems. J. Eng. Struct. 85, 1–10. doi: 10.1016/j.engstruct.2014.12.007

CrossRef Full Text | Google Scholar

Charalampakis, A. E., and Dimou, C. K. (2010). Identification of Bouc–Wen hysteretic systems using particle swarm optimization. J. Comput. Struct. 88, 1197–1205. doi: x.1016/j.compstruc.2010.06.009

CrossRef Full Text | Google Scholar

Charalampakis, A. Eastward., and Koumousis, V. M. (2008a). Identification of Bouc–Wen hysteretic systems by a hybrid evolutionary algorithm. J. Sound Vib. 314, 571–585. doi: 10.1016/j.jsv.2008.01.018

CrossRef Total Text | Google Scholar

Charalampakis, A. E., and Koumousis, V. K. (2008b). On the response and dissipated energy of Bouc–Wen hysteretic model. J. Sound Vib. 309, 887–895. doi: 10.1016/j.jsv.2007.07.080

CrossRef Total Text | Google Scholar

Chatzi, Due east. N., and Smyth, A. W. (2009). The unscented Kalman filter and particle filter methods for nonlinear structural system identification with non-collocated heterogeneous sensing. J. Struct. Control Hlth. Monit. 16, 99–123. doi: 10.1002/stc.290

CrossRef Full Text | Google Scholar

Chatzi, E. N., Smyth, A. W., and Masri, Southward. F. (2010). Experimental application of on-line parametric identification for nonlinear hysteretic systems with model uncertainty. J. Struct. Saf. 32, 326–337. doi: x.1016/j.strusafe.2010.03.008

CrossRef Total Text | Google Scholar

Ching, J., and Chen, Y. C. (2007). Transitional Markov Chain Monte Carlo method for Bayesian model updating, model form option, and model averaging. J. Eng. Mech. 133, 816–832. doi: ten.1061/(ASCE)0733-9399(2007)133:7(816)

CrossRef Full Text | Google Scholar

Constantinou, M. C., Soong, T. T., and Dargush, G. F. (1998). Passive Energy Dissipation Systems for Structural Design and Retrofit. Multidisciplinary Centre for Earthquake Engineering Research.

Deb, Chiliad., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE J. Trans. Evol. Comput. 6, 182–197. doi: 10.1109/4235.996017

CrossRef Full Text

Erazo, K., and Nagarajaiah, S. (2018). Bayesian structural identification of a hysteretic negative stiffness earthquake protection organisation using unscented Kalman filtering. J. Struct. Command Hlth. Monit. 25, one–xviii. doi: 10.1002/stc.2203

CrossRef Full Text | Google Scholar

Foliente, G. C. (1995). Hysteresis modeling of wood joints and structural systems. J. Struct. Eng. 121, 1013–1022. doi: 10.1061/(ASCE)0733-9445(1995)121:half dozen(1013)

CrossRef Total Text | Google Scholar

Green, P. L. (2015). Bayesian system identification of dynamical systems using big sets of grooming data: a MCMC solution. J. Probabilist. Eng. Mech. 42, 54–63. doi: x.1016/j.probengmech.2015.09.010

CrossRef Full Text | Google Scholar

Green, P. 50., Cross, E. J., and Worden, K. (2015). Bayesian arrangement identification of dynamical systems using highly informative training data. J. Mech. Syst. Indicate PR 56, 109–122. doi: 10.1016/j.ymssp.2014.ten.003

CrossRef Full Text | Google Scholar

Hornig, Thousand. H., and Flowers, M. T. (2005). Parameter label of the Bouc-Wen mechanical hysteresis model for sandwich blended materials using real coded genetic algorithms. Int. J. Acoust. Vib. ten:7381. doi: ten.20855/ijav.2005.10.2176

CrossRef Full Text | Google Scholar

Ikhouane, F., and Gomis-Bellmunt, O. (2008). A limit bicycle arroyo for the parametric identification of hysteretic systems. J. Syst. Control Lett. 57, 663–669. doi: 10.1016/j.sysconle.2008.01.003

CrossRef Full Text | Google Scholar

Ikhouane, F., Mañosa, V., and Rodellar, J. (2007). Dynamic properties of the hysteretic Bouc-Wen model. J. Syst. Control Lett. 56, 197–205. doi: 10.1016/j.sysconle.2006.09.001

CrossRef Full Text | Google Scholar

Ikhouane, F., and Rodellar, J. (2007). Systems With Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model. Hong Kong: John Wiley & Sons.

Google Scholar

Jeen-Shang, L., and Yigong, Z. (1994). Nonlinear structural identification using extended Kalman filter. J. Comput. Struct. 52, 757–764.

Google Scholar

Kontoroupi, T., and Smyth, A. West. (2017). Online Bayesian model assessment using nonlinear filters. J. Struct. Control Hlth. Monit. 24, 1–xv. doi: 10.1002/stc.1880

CrossRef Full Text | Google Scholar

Lei, Y., and Jiang, Y. Q. (2011). "A two-phase Kalman estimation arroyo for the identification of structural parameters under unknown inputs," in The Twelfth East asia-Pacific Conference on Structural Engineering and Construction, 3088–3094.

Google Scholar

Ma, F., Zhang, H., Bockstedte, A., Foliente, Thousand. C., and Paevere, P. (2004). Parameter assay of the differential model of hysteresis. J. Appl. Mech. 71, 342–349. doi: 10.1115/1.1668082

CrossRef Total Text | Google Scholar

Masri, Southward. F., Chassiakos, A. 1000., and Caughey, T. One thousand. (1992). Structure-unknown non-linear dynamic systems: identification through neural networks. J. Smart Mater. Struct. ane, 45–56. doi: ten.1088/0964-1726/1/ane/007

CrossRef Full Text | Google Scholar

Monti, G., Quaranta, G., and Marano, G. C. (2009). Genetic-algorithm-based strategies for dynamic identification of nonlinear systems with dissonance-corrupted response. J. Comput. Civil Eng. 24, 173–187. doi: 10.1061/(ASCE)CP.1943-5487.0000024

CrossRef Full Text | Google Scholar

Mu, T., Zhou, 50., and Yang, J. N. (2013). Comparison of adaptive structural impairment identification techniques in nonlinear hysteretic vibration isolation systems. J. Earthq. Eng. Eng. Vib. 12, 659–667. doi: 10.1007/s11803-013-0204-y

CrossRef Full Text | Google Scholar

Muller, O., Savino, F., Rubinstein, One thousand., and Foschi, R. O. (2012). "Performance-based seismic design: a search-based cost optimization with minimum reliability constraints," in Structural Seismic Blueprint Optimization and Earthquake Engineering: Formulations and Applications 23–50. doi: 10.4018/978-i-4666-1640-0.ch002

CrossRef Total Text

Muto, G., and Brook, J. Fifty. (2008). Bayesian updating and model class pick for hysteretic structural models using stochastic simulation. J. Vib. Command 14, 7–34. doi: x.1177/1077546307079400

CrossRef Full Text | Google Scholar

Muto, Thousand. G. (2007). Application of Stochastic Simulation Methods to System Identification. Ph.D., California Institute of Technology.

Noori, K. (1984). Random Vibration of Degrading Systems With General Hysteretic Behavior. Ph.D., University of Virginia.

Noori, G., Wang, H., Altabey, W. A., and Silik, A. I. H. (2018). A modified wavelet energy rate based harm identification method for steel bridges. Int. J. Sci. Technol. doi: 10.24200/sci.2018.20736

CrossRef Full Text

Ortiz, G. A., Alvarez, D. A., and Bedoya-Ruíz, D. (2013). Identification of Bouc-Wen type models using multi-objective optimization algorithms. J. Comput. Struct. 114, 121–132. doi: ten.1016/j.compstruc.2012.ten.016

CrossRef Full Text | Google Scholar

Ortiz, G. A., Alvarez, D. A., and Bedoya-Ruíz, D. (2015). Identification of Bouc–Wen type models using the Transitional Markov chain Monte Carlo method. J. Comput. Struct. 146, 252–269. doi: x.1016/j.compstruc.2014.x.012

CrossRef Full Text | Google Scholar

Park, Y. J., Wen, Y. Chiliad., and Ang, A. (1986). Random vibration of hysteretic systems under bi-directional ground motions. Earthq. Eng. Struct. D xiv, 543–557. doi: 10.1002/eqe.4290140405

CrossRef Full Text | Google Scholar

Peng, G. R., Li, Westward. H., Du, H., Dengc, H. X., and Alicia, Grand. (2014). Modelling and identifying the parameters of a magneto-rheological damper with a force-lag phenomenon. J. Appl. Math. Model. 38, 3763–3773. doi: 10.1016/j.apm.2013.12.006

CrossRef Full Text | Google Scholar

Puttige, 5. R., and Anavatti, S. G. (2007). "Comparison of real-time online and offline neural network models for a UAV," in IEEE International Joint Conference on Neural Networks (Republic of hungary), 412–417.

Google Scholar

Saadat, S., Buckner, G. D., Furukawa, T., and Noori, One thousand. Due north. (2003). "Nonlinear system identification of base-excited structures using an intelligent parameter varying (IPV) modeling approach," in Proceedings of Smart Structures and Materials, International Society for Optics and Photonics (San Diego, CA), 555–564.

Google Scholar

Saadat, Southward., Buckner, G. D., Furukawa, T., and Noori, Thou. N. (2004a). An intelligent parameter varying (IPV) approach for not-linear system identification of base excited structures. Int. J. Nonlin. Mech. 39, 993–1004. doi: 10.1016/S0020-7462(03)00091-Ten

CrossRef Full Text | Google Scholar

Saadat, Due south., Buckner, G. D., and Noori, Thousand. N. (2007). Structural system identification and damage detection using the intelligent parameter varying technique: an experimental written report. J. Struct. Health Monit. 6, 231–243. doi: 10.1177/1475921707081980

CrossRef Full Text | Google Scholar

Saadat, S., Noori, M. North., Buckner, Grand. D., Furukawa, T., and Suzuki, Y. (2004b). Structural health monitoring and damage detection using an intelligent parameter varying (IPV) technique. Int. J. Nonlin. Mech. 39, 1687–1697. doi: ten.1016/j.ijnonlinmec.2004.03.001

CrossRef Total Text | Google Scholar

Sohn, H. (1998). A Bayesian Probabilistic Approach to Impairment Detection for Ceremonious Structures. Ph.D., Stanford university.

Soong, T. T., and Spencer, B. F. (2002). Supplemental free energy dissipation: state-of-the-art and state-of-the-practice. Eng. Struct. 24, 243–259. doi: x.1016/S0141-0296(01)00092-X

CrossRef Total Text | Google Scholar

Spencer, B. F. Jr, and Nagarajaiah, Southward. (2003). Land of the fine art of structural control. J. Struct. Eng. 129, 845–856. doi: 10.1061/(ASCE)0733-9445(2003)129:7(845)

CrossRef Full Text

Wen, Y. K. (1975). Guess method for nonlinear random vibration. J. Eng. Mech. 101, 389–401.

Google Scholar

Wen, Y. Grand. (1976). Method for random vibration of hysteretic systems. J. Eng. Mech. 102, 249–263.

Google Scholar

Wen, Y. G. (1980). Equivalent linearization for hysteretic systems nether random excitation. J. Appl. Mech. 47, 150–154. doi: ten.1115/1.3153594

CrossRef Full Text | Google Scholar

Wen, Y. M. (1986). Stochastic response and damage analysis of inelastic structures. Probabilist. Eng. Mech. ane, 49–57. doi: ten.1016/0266-8920(86)90009-3

CrossRef Full Text | Google Scholar

Wen, Y. K. (1989). Methods of random vibration for inelastic structures. J. Appl. Mech. Rev. 42, 39–52. doi: ten.1115/1.3152420

CrossRef Full Text | Google Scholar

Wen, Y. One thousand., and Yeh, C. H. (1989). Biaxial and torsional response of inelastic structures under random excitation. J. Struct. Saf. 6, 137–152. doi: 10.1016/0167-4730(89)90016-seven

CrossRef Full Text | Google Scholar

Worden, 1000., and Hensman, J. J. (2012). Parameter estimation and model choice for a class of hysteretic systems using Bayesian inference. Mech. Syst. Signal PR 32, 153–169. doi: x.1016/j.ymssp.2012.03.019

CrossRef Full Text | Google Scholar

Wu, M., and Smyth, A. W. (2007). Application of the unscented Kalman filter for existent-time nonlinear structural organization identification. J. Struct. Control Hlth. Monit. fourteen, 971–990. doi: 10.1002/stc.186

CrossRef Full Text | Google Scholar

Wu, G., and Smyth, A. W. (2008). Real-fourth dimension parameter interpretation for degrading and pinching hysteretic models. Int. J. Nonl. Mech. 43, 822–833. doi: 10.1016/j.ijnonlinmec.2008.05.010

CrossRef Full Text | Google Scholar

Yang, J. Northward., Lin, S., Huang, H., and Zhou, L. (2006). An adaptive extended Kalman filter for structural impairment identification. J. Struct. Control Hlth. Monit. thirteen, 849–867. doi: 10.1002/stc.84

CrossRef Full Text | Google Scholar

Yuen, K. Five., and Katafygiotis, 50. South. (2001). Bayesian time–domain approach for modal updating using ambience data. Probabilist. Eng. Mech. sixteen, 219–231. doi: 10.1016/S0266-8920(01)00004-2

CrossRef Full Text | Google Scholar

Zhang, B. T., and Cho, D. Y. (2001). Organization identification using evolutionary Markov Concatenation Monte Carlo. J. Syst. Architect. 47, 587–599. doi: 10.1016/S1383-7621(01)00017-0

CrossRef Full Text | Google Scholar

Zhao, Y., Noori, M., and Altabey, Westward. A. (2017a). Harm detection for a beam under transient excitation via three different algorithms. J. Struct. Eng. Mech. 63, 803–817.

Google Scholar

Zhao, Y., Noori, M., Altabey, W. A., and Bahram Beheshti-Aval, B. (2017b). Mode shape based damage identification for a reinforced concrete beam using wavelet coefficient differences and multi-resolution analysis. J. Struct. Control Wellness Monitor. 25, i–41. doi: x.1002/stc.2041

CrossRef Full Text

Zhao, Y., Noori, M., Altabey, W. A., and Wu, Z. (2018). Fatigue damage identification for blended pipeline systems using electrical capacitance sensors. J. Smart Mater. Struct. 27:8. doi: 10.1088/1361-665X/aacc99

CrossRef Full Text | Google Scholar

Zheng, Due west., and Yu, Y. (2013). Bayesian probabilistic framework for damage identification of steel truss bridges under articulation uncertainties. Adv. Civil Eng. 2013:307171. doi: 10.1155/2013/307171

CrossRef Total Text | Google Scholar

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