Nonlinear Identification of a ... - Wiley Online Library

Computer-Aided Civil and Infrastructure Engineering 29 (2014) 221–233

Nonlinear Identification of a Magneto-Rheological Damper Based on Dynamic Neural Networks Marzuki Khalid, Rubiyah Yusof,∗ Majid Joshani & Hazlina Selamat Centre for Artificial Intelligence and Robotics, Universiti Teknologi Malaysia, Kuala Lumpur 54100, Malaysia

& Mohamad Joshani Malaysia-Japan International Institute of Technology, Universiti Teknologi Malaysia, Kuala Lumpur 54100, Malaysia

Abstract: Semi-active control of dynamic response of civil structures with magneto-rheological (MR) fluid dampers has emerged as a novel revolutionary technology in recent years for designing “smart structures.” A small-scale MR damper model with the valve mode mechanism has been examined in this research using dynamic recurrent neural network modeling approach to reproduce its hysteretic nonlinear behavior. Modified Bouc–Wen model based on nonlinear differential equations has not only been employed as the reference model to provide a comprehensive training data for the neural network but also for comparison purposes. A novel frequency and amplitude varying displacement input signal (modulated chirp signal) associated with a random supply voltage has been introduced for persistent excitation of the damper in such a way to cover almost all of its operating conditions. Finally a series of validation tests were conducted on the proposed model which proved the appropriate performance of the model in terms of accuracy and capability for realization.

1 INTRODUCTION Semi-active vibration control systems have been introduced during the last two decades as a solution to rectify the nonadaptability problems of passive control systems. It can also be used as a solution ∗ To whom correspondence should be addressed. E-mail: rubiyah@ic. utm.my.

C 2013 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/mice.12005

for unreliability problems and high-power requirement of active vibration control systems. Because the concept of semi-active control systems has been introduced in the vibration control engineering discipline, several devices with various vibration dampening mechanisms have been developed (e.g., semi-active tuned liquid column dampers with servo valve mechanism, semi-active friction dampers with piezo-actuation mechanism, semi-active sloshing tank with controllable blades, semi-active electro-rheological and magnetorheological dampers, etc.) to improve the performance of the previous passive and active vibration control systems by means of an integrated version, which includes all the advantages of the aforementioned systems. Moreover, in recent years a great deal of research has been intensively devoted to developing control algorithms (strategies) to tackle the dynamic response of structural or mechanical systems that are equipped with such innovative devices. A proper integration of semiactive damping devices with suitable control schemes will render a high-performance vibration mitigation system that can be utilized efficiently in a wide range of applications such as structural vibration control problems (e.g., cable-stayed bridges, high-rise buildings, base isolation systems) and also in mechanical vibration suppression systems (such as vehicle and train suspension systems, landing gears, robotic joints, or even biomedical cases such as prosthetic limbs). Therefore, semiactive control technology enables engineers of the 21st century to design smart (intelligent or adaptive) flexible civil structures with highest efficiency in terms of economy, minimal structural skeleton weight, maximal

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lateral resistance, and minimal structural damage after a severe earthquake disaster. In this context, a magneto-rheological (MR) damper is considered as one of the most promising supplemental smart damping devices for semi-active control of structures. It utilizes smart materials of a high level of accuracy and responsiveness toward instantaneous changes of controllable variables (Spencer et al., 1997; Dyke et al., 1998; Cheng et al., 2008). Nowadays, various flexible civil structures, which are seismically excited or prone to high winds, can be equipped with such smart damper as an interconnector of flooring systems or as a based isolator or even as an interconnector of adjacent buildings in sky bridges to dampen the system (Hiemenz et al., 2003; Johnson ´ et al., 2003; Duan et al., 2005; Cundumi and Suarez, 2008; Kim et al., 2010; Bitaraf et al., 2012). MR dampers produce variable damping by changing continuously and swiftly the rheological properties of a particular type of fluid (MR fluid) through the alteration of the magnitude of the applied magnetic field (Guglielmino et al., 2008). With the availability of modern designs, MR dampers deliver a considerable set of benefits such as large force capacity, low power requirements, high robustness, cost effectiveness, and excellent performance over a variety of operating conditions (Bossis et al., 2003; Guglielmino et al., 2008). In addition, they are reliable and possess fail-safe mode feature that guarantees their minimal acceptable resistance properties in hazardous situations such as during earthquakes, where there are interruptions in the power supply (Chu et al., 2005; Kwok et al., 2006). In contrast to the active control devices, being semiactive, MR dampers do not have the capability to destabilize the structure due to the similarity of their performance to braking systems rather than the actuator systems. This is because they can only exert dissipative forces to remove the destructive energy from the structure although the active control devices (actuators) have the potential to destabilize the structure by injecting external energy to the system. The controller in the semi-active control system produces control signals that modify the damping characteristics of the MR damper and subsequently the damping properties of the primary structure whereas the controller in the active control system sends signals to the actuator to exert huge external forces to the structure (Chu et al., 2005; Ikhouane and Rodellar, 2007). It must be mentioned here that there are three types of MR dampers, namely, the shear mode, the squeeze mode, and the valve mode. The operating mechanism of the shear mode MR damper is based on the relative

inter-laminar displacement of two parallel plates that are separated by a thin layer of MR fluid. The controllable MR fluid can adjustably resist against the displacement of the upper plate with respect to the lower plate according to the magnitude of the exerted input voltage. The applied voltage will align the chains of the magnetic particles perpendicular to the surface of the MR fluid layer and the two plates that prevent the sliding of upper plate. The squeeze mode MR damper also consists of two parallel plates with an interlayer MR fluid that resist against moving toward each other in the presence of magnetic field. The adjustable voltage input signal will determine the compressive force that the damper can produce in a squeeze mode. The valve mode MR damper, which is applicable in large-scale civil structures for its ability to offer a high force capacity, works on the principle of valving mechanism. It consists of a piston rod with embedded solenoid that can travel inside a cylinder full of MR fluid. The valve formed by the distance between the central piston and the peripheral cylinder can become fully or partially closed in an adjustable manner according to the intensity of the applied input voltage signal. The higher the voltage, the stronger the magnetic field inside the chamber, and the magnetic particles are aligned with higher strength in the direction of magnetic flux, perpendicular to the MR fluids’ flow. Hence, the energy dissipation rate in this adjustable valve is increased when this voltage is increased and vice versa. In spite of all the promising features of MR dampers in structural control applications, there is a challenging problem in their modeling and controller design due to the complexity of their dynamics. The MR damper has the potential to give significant effects on the structural performance with only a small change in the control signal. This is caused by the presence of the large hysteresis in the characteristics of an MR damper that will require a rather robust and reliable control strategy (Guglielmino et al., 2008). Because hysteresis is a strongly nonlinear physical phenomenon, linearization is not able to encapsulate the observed behavior of the hysteretic systems (Bertotti and Mayergoyz, 2006). Much research has been performed to describe the hysteretic behaviors based on general mathematical models (Macki et al., 1993; Visintin, 1994; Bouc and Boussa, 2002; Mayergoyz, 2003). Approaches for modeling and characterization of the dynamics of MR dampers can be categorized into analytical and numerical procedures. The analytical method is a straightforward approach in which the parameters of an actual system model are used directly to represent physical quantities, which is particularly effective for linear time invariant systems (Jiang et al.,

Nonlinear identification of a magneto-rheological damper

2007). In the numerical approach, the input–output map is characterized and determined by a system model that may not have any explicit physical meaning. However, this method does not require complete measurement of the systems’ internal states and has a better adaptability than the analytical method, particularly for complicated nonlinear plants (Jiang et al., 2007). In this study, the modified Bouc–Wen (B-W) model has been utilized as a reference analytical model for generating the required training data set for constructing the proposed numerical model. The B-W model, which is a smoothly varying differential hysteresis model and is suitable in modeling the hysteretic phenomena, was introduced in Bouc (1967) and generalized by Wen (1976) in a form that can be used to reproduce a variety of hysteretic patterns (Dobson et al., 1997). The B-W model was utilized in modeling the MR dampers very successfully except for the lack of precision in emulating the roll-off effect in the near zero velocities when the acceleration and velocity have opposite signs (Dyke et al., 1996; Spencer et al., 1997). To capture this phenomenon, observed in experimental tests, the same authors proposed a phenomenological model (modified B-W model) (Spencer et al., 1997). They added a viscous damping element to a simple B-W model to create a nonlinear roll-off effect in the MR damper and a spring element to create an accumulator stiffness. Although this method delivers a precise emulation of the MR damper, it requires 14 unknown parameters of the model to be determined through experimental tests. The past few decades witnessed a rapid growth in the use of artificial intelligence (AI) techniques in the industry and science for different purposes such as control, modeling, forecasting, etc. (Wang, 1994; Mills et al., 1995; Ruano, 2005). The application of artificial neural network (ANN) techniques in civil engineering was introduced for the first time by Adeli and Yeh (1989). Since then, ANN techniques have been employed in the civil engineering discipline for optimization, control, identification, and forecasting problems. For instance, this powerful technique has been utilized in the fields of structural identification and health monitoring (Jiang and Adeli, 2005; Adeli and Jiang, 2006; Lam et al., 2006; Jiang et al., 2007), structural control (Kim and Roschke, 2006; Yuen et al., 2007; Jiang and Adeli, 2008a, b), structural optimization (Park and Adeli, 1995; Park and Adeli, 1997; Graf et al., 2012), earthquake prediction (Adeli and Panakkat, 2009; Panakkat and Adeli, 2009), transportation engineering (Adeli and Samant, 2000; Dharia and Adeli, 2003; Ghosh-Dastidar and Adeli, 2003; Jiang, 2003), and construction (Cruz and Marques, 2012). Also, recently several numerical models based on various AI techniques such as ANN, fuzzy logic, neuro-

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fuzzy, and wavelets have been proposed for modeling the MR dampers. An ANN model has been developed to emulate the dynamics of the MR damper in Chang and Roschke (1998). Schurter and Roschke (2000) modeled an MR damper using Takagi–Sugeno– Kang fuzzy inference system using the MR damper model developed in Spencer et al. (1997) as the reference model. A forward neuro-fuzzy model has also been developed for a heavy duty large-scale 30-tonne MR damper (Schurter and Roschke, 2000; Peschel and Roschke, 2001; Sunakoda, 2004). Also, based on an exhaustive series of experimental tests on a small scale MR damper, a fuzzy model was trained and adapted for the device using adaptive neuro-fuzzy inference technique for vibration control of a railcar (Atray and Roschke, 2004). In Wang and Liao (2005), a forward MR damper model consisting of a recurrent neural network (RNN) was presented. A Takagi–Sugeno fuzzy identifier was also used in modeling the MR damper for its control purposes (Du and Zhang, 2009). However, most of the models obtained did not explore various ranges of the operating modes and ranges of the MR dampers. Most of the ANN models behave as a black box model. The model depends on the set of training data used in obtaining it. This type of model is more suitable for interpolation problems rather than extrapolation problems. Thus, the selection of appropriate training data that can trace all the operating modes and range of a device is of fundamental importance in ensuring the accuracy of the ANN model. One of the ways to do this is to excite the system to be modeled with suitable input signals. In this article, a combination of frequency varying sinusoidal (chirp) signal with amplitude modulation and random Gaussian signals to excite the device over its different working ranges is proposed. In this work, the nonlinear modified B-W model proposed in Spencer et al. (1997) has been used as the reference model to generate the training data to produce the ANN-based model as it is highly accurate. Several tests are done to evaluate the robustness of the modeling technique by varying the parameters such as displacement amplitude, frequency, etc., covering several spectrums of the operating range of the device. This article is organized as follows: in Section 2, the behavior of MR dampers and their relevant B-W mathematical equations are discussed. In Section 3, the proposed modeling methodology employed in this research is introduced. Subsequently, in Section 4 the efficiency of such identification scheme is demonstrated and discussed using appropriate validation tests and their results. Finally, the conclusions in Section 5 end the article.

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2 MR DAMPERS DYNAMICS AND BEHAVIOR MR fluid is a noncolloidal suspension that consists of micron-sized magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil (Dyke et al., 1996). The unique feature of such kind of fluid is its ability to instantaneously change its rheological properties once a magnetic field (by means of a solenoid embedded inside the damper) is introduced to the system. Therefore, the transformation of MR fluids from free-flowing linear viscous liquid state to semisolid state with exhibition of viscoplastic hysteretic behavior is possible by introducing a magnetic field to the chamber of fluid (Carlson and Weiss, 1994; Ashour et al., 1996; Bossis et al., 2003; Sodeyama et al., 2003). Therefore, continuous variable damping can be achieved with no moving mechanical parts other than the piston, which gives MR dampers better reliability and simplicity in comparison with other semi-active damper systems (Chu et al., 2005). The use of the variable viscous material in the valving mechanism of MR dampers allows the control of the solenoid current using only low-voltage supply such as the conventional batteries (Guglielmino et al., 2008). Moreover, MR fluids can operate efficiently in temperature ranges from –40 up to 150 degrees Celsius with merely a small change in their yield stress (Carlson and Weiss, 1994). The small-scale MR damper model used in this study is based on a prototype RD-1005-3 MR damper, which is 21.6 cm long in its extended position, 3.81 cm in diameter, and has a stroke of 2.54 cm (Spencer et al., 1997; Zapateiro De La Hoz, 2009). This MR damper, with the valve mode mechanism, can generate a force up to 3,000 N using only a small battery with a capacity of less than 10 W (Kim, 2007). As mentioned earlier, the hysteretic behavior plays the major role in the performance of MR dampers. Therefore, in modeling such devices, a profound understanding of hysteresis phenomenon is needed. Generally, hysteresis can be characterized as a special kind of memory-based relationship among input and output signals (Macki et al., 1993). In other words, hysteresis can be interpreted as a type of nonlinearity with a memory feature (history dependent) that shows itself through branching (Mayergoyz, 2003). Furthermore, the area enclosed by a hysteresis loop is a measure of the amount of energy dissipated per one cycle of input oscillation (Guglielmino et al., 2008). Spencer et al. (1997) proposed an improved model of MR dampers by adding a parallel spring and a tandem dashpot element with the previous simple B-W model to take into account the round-off effect in low velocities and the accumulator stiffness. It is important to note that the spring-dashpot model added to the original

Table 1 Parameters of modified B-W model for prototype RD-1005 MR damper Parameter x0 C0a C0b C1a C1b k0 k1

Value

Parameter

Value

0m 21 Ns/cm 3.50 Ns/cmV 283 Ns/cm 2.95 Ns/cmV 46.90 N/cm 5 N/cm

αa αb β γ A n η

140 N/cm 695 N/cmV 363 cm−2 363 cm−2 301 2 190 s−1

B-W model in Wen (1976) aims to create the linear visco-elastic behavior of the MR damper only and it is the B-W model that generates the hysteretic behavior. Considering x, x, ˙ v, and fmr as the dampers’ displacement, velocity, supplied voltage, and output force, respectively, the modified B-W model utilized as the reference model in this study is described by the following equations: fmr = αz + c0 (x˙ − y˙ ) + k0 (x − y) + k1 (x − x0 ) (1) fmr = c1 y˙ + k1 (x − x0 ) y˙ =

1 {αz + c0 x˙ + k0 (x − y)} c0 + c1

(2) (3)

z˙ = −γ |x˙ − y˙ |z|z|n−1 − β(x˙ − y˙ )|z|n + A(x˙ − y˙ ) (4) u˙ = −η(u − v)

(5)

In Equations (1) to (4), α, β, γ , A, n are the parameters that control the scale and shape of the hysteresis loops and z is an internal variable that defines the hysteresis components of the output force according to the displacement and velocity of the damper. Parameter y denotes the displacement for the added spring-dashpot elements to the conventional B-W model. The parameter u is a filtered version of the supplied voltage tuned by the η parameter. The coefficients for hysteretic component, velocity, and displacement denoted by α, (c0 , c1 ), (k0 , k1 ), respectively, in Equations (1) and (2), are linear variables of u as described in Equations (6)–(8). α = α(u) = αa + αbu

(6)

c0 = c0 (u) = c0a + c0bu

(7)

c1 = c1 (u) = c1a + c1bu

(8)

The parameters of the modified B-W model used in this study are shown in Table 1 (Dyke et al., 1996;


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Spencer et al., 1997). As presented in this table, the modified B-W model can be defined by 14 constants. In Spencer et al. (1997), these constants were tuned using the least squares optimization method based on experimental data.

3 DYNAMIC ANN MODELING APPROACH ANN models can be trained to represent many types of systems and they have better advantages and are more accurate when compared to traditional models such as linear models, logarithmic models, exponential regression models, power regression models, and even complicated inference systems such as fuzzy logic models. In fact, some of these techniques can be considered as a specific class of ANNs. For example, fuzzy logic systems can be categorized as a special type of neural networks, referred to as the radial basis function neural networks. At the same time, ANNs possess the major advantage of having the capability of parallel processing in each of their layers. This capability can enhance the computational performance of an ANN when used as an identifier or a controller. This is one of the main reasons for the selection of ANNs for solving MR dampers identification problem. Therefore, such models can be applied in real-time control systems with time constraints for model processing. In this section, the use of RNN in modeling the MR damper is proposed. Dynamic neural network structures are designed by a certain time-delayed input vector to consider the memory of past information (Adeli and Jiang, 2006). Some of the other texts have referred to these network structures as tapped-delay-line networks, neural autoregressive with exogenous inputs, or neural autoregressive models (Adeli and Jiang, 2009). The training of a dynamic network is substantially more complicated and time consuming than the training of a conventional network, because in the former, both the input and output signals are not single valued but in the form of time series (Jiang and Adeli, 2005). However, when the input data are time-dependent, it allows the function approximation problem to be solved using the current and past information for the prediction of future behavior of the system (Graf et al., 2010). Therefore, the system model used in this study receives not only the current states of the device, but also their delayed values as the input signals in the function approximation process. Equation (9) is called the “regression” equation that describes the system to be modeled in terms of the measured input and output signals as well as their delayed versions (Mills et al., 1995). It allows the vector of the system parameters, θ , to be estimated using only the input

Fig. 1. The architecture of tapped delay RNN system with R inputs and S2 outputs.

signals fed into the system and the corresponding output signals. y(k) = g(θ, y(k − 1), . . . , y(k − na ), u(k − 1), . . . , u(k − nb))

(9)

In the equation, na and nb are the numbers of input and output delays required by the autoregressive model, respectively. ANN is also an autoregressive platform that can be optimized, tuned and used as models to describe various signals. The RNN is a dynamic type of network for its recursive structure that receives delayed signals as complementary time series inputs. As shown in Figure 1, by using RNN, a network structure with feedback links that enables the current and past signals to travel around the loops is constructed. The system consists of an R-dimensional input with the number of tapped delays, d. This produces an (R×d) input system that has an arbitrary S2 output. The block that is labeled “TD” in Figure 1, collects d delayed input signals of the system. These inputs will propagate through the input layer weights (“IW”) toward the network’s hidden layer that has a transition function “f 1 ” and S1 number of neurons. The hidden layer neurons are connected to the output layer through the output layer weights (“LW”). The outputs of the hidden neurons are also feedback via a delayed feedback (“D”) and LW. The weights in this feedback loop allow the network to reconstruct the relationship between the current and previous states of signals. The bias blocks (“b1 ” and “b2 ”) shift the output of neurons by a constant value and finally a set of transition functions in the output layer (“f2 ”) maps the data to S2 outputs. For the RNN model, θ denoted in Equation (9) consists of the input and output layers’ weights, biases, and transition functions, as shown in Equation (10). θ = [IW; LW; b1 ; b2 ; f1 ; f2 ]

(10)

Generally, there are two different strategies for modeling nonlinear hysteretic systems. The first one is based on the segmentation (partitioning) of data to separate

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the hysteresis branches. However, this approach requires at least two models to be individually trained and tested. Furthermore, the problem of nonunique forecasting outputs may be encountered in the boundary regions of the separated branches of a hysteretic system (Basheer, 2000). The second approach in modeling nonlinear hysteretic systems involves the use of a single identifier receiving certain input data that can define the characteristics of the system model. In this research, although the acceleration sign can be defined as ascending or descending trend for the output force of an MR damper, a single RNN system is used to model the MR damper receiving the current and past velocity of the device that can determine the accelerations sign for the network internally. In this way, the whole available data from the MR damper can be used to construct and train a single network as a model of the device. In the following section, the generation of simulated data and network training procedures are described. 3.1 Data generation and preprocessing Models used in structural control systems must be able to reconstruct the dynamics of the MR damper for different displacements and velocities. Most unsuccessful modeling scenarios suffer from lack of knowledge on the system variables or insufficient training data, which will cause an unexpected extrapolation in the model performance. It must be noted that although a model may give a good approximation of a hysteresis loop for a certain input excitation, it may not necessarily be appropriate to represent the behavior of the hysteretic system under other types of input excitations. This issue has been considered in the identification of the modified B-W MR damper model discussed in this article. This problem is addressed by designing the proposed system model with the device being tested under comprehensive persistently exciting input signals (displacement, velocity, and voltage). Persistently exciting input signal is a sufficiently rich input signal used in estimating plant parameters to excite all modes of the plant so that the parameter estimates converge to the true values. This is a very important condition for convergence of the estimated parameters in system identification literature and has been addressed, for example, in Goodwin and Sin (1984) and Wan et al. (2001), and applied in the identification process of various civil engineering systems such as the open channel water flow systems (Diamatis et al., 2011). The variations of the input signals used in this study are provided in Table 2. To feed proper stimulation input signals to the MR damper to capture its different operating ranges, a comprehensive variation of each variable in Table 2 is

Table 2 Input variables chosen for the model and their ranges of variations Variable Displacement Velocity (or displacement frequency) Supplied voltage

Variation range −0.03 m to 0.03 m 0.5 (Hz) to 5 (Hz) 0 (V) to 5 (V)

required. This is done by varying one of the variables while keeping the other two variables at certain constant values. This, however, will cause a large amount of data to be processed and will result in longer simulation time for B-W model or in the case of real-time MR damper experiment, will require a heavier data acquisition process. In this study, a random variation of supplied voltage is accompanied by ordered variations of displacement and velocity to trace almost all of the operating range of the MR damper. A chirp signal, which is a sinusoidal wave with a linear increment in the frequency, is applied to sweep a desired range in frequency of displacement variations. At the same time, another chirp signal is modulated on the displacement waveform to generate sinusoidal variations of displacement amplitudes. These variations of displacement, amplitude, and frequency are repeated five times although in each time a certain range of supply voltage is experienced by the damper randomly. The displacement variation is produced using Equation (11): (11) x(t) = sin(2π f2 .t) ∗ sin(2π f1 .t) where f 1 is the frequency of the main sinusoidal displacement that varies linearly from 0.5 Hz to 5 Hz and f 2 is the frequency of the displacement amplitude modulated on the main sinusoidal displacement signal and varies between 0.03 Hz to 0.3 Hz. In both cases, 20 different frequencies are used for each range of frequencies. The input and output training data obtained from the modified B-W MR damper model are shown in Figures 2 and 3, respectively. Figure 2 shows the displacement, velocity, and supply voltage to the MR damper, whereas Figure 3 shows the output force generated. The force–displacement and force–velocity variations are illustrated in Figures 4a and b. It can be observed that the chosen input data could successfully cover most of the working space (region) of the MR damper in force–displacement and force–velocity planes. The B-W model which is a nonlinear differential equation was solved using ODE45 adaptive variable time sampling solver (Dormand–Prince) available in MATLAB software to avoid the bifurcation effects in


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Fig. 2. Excitation inputs for training RNN model (the training data).

Fig. 4. (a) Force versus displacement and (b) force versus velocity spaces traced by the provided training data.

Fig. 3. Output force generated due to training excitations over a 30-second time interval.

system outputs which might occur using a fixed time step size solver. The input–output data for the system are resampled with a fixed time step size of 0.001 seconds to redistribute the data uniformly for the whole simulation interval. In this way, the effectiveness of different ranges of data on the RNN model will be obtained. 3.2 Model design In this section, the simulated data obtained from the modified B-W model are used to obtain the neural network model of the MR damper using the RNN platform. The most significant parameters that affect the output force of the MR damper are the displacement, velocity, acceleration, and supply voltage. Therefore, these variables should be defined as inputs of the model. At the same time a delayed version of these

signals will be fed to the network to allow it to recognize the system’s dynamics caused by these parameter variations. Because the delayed version of the velocity data contains the information about the acceleration, the acceleration is not included in the input data sets; hence, the model will receive three signals and their delayed versions and these are used in reconstructing output force of the damper. Therefore, the model can be classified as a multi-input–single output process. After several simulation runs performed, to get an acceptable accuracy and a reasonable amount of computational effort to obtain the RNN model for the MR damper, the number of neurons in the hidden layer of the neural network has been chosen to be 50. The input layers consist of the three signals as mentioned with five delayed versions of each signal. In the second stage shown in Figure 5, a back propagation Levenberg–Marquardt optimization function is employed in the adaptation of the model with the simulated training data. To avoid the use of the same training data for validating the optimized network, the simulated data were divided into two portions: 70% were used for the training process and 30% were used for testing and validating the training procedure. The network optimization

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Fig. 5. Block diagram for tapped delay RNN modeling of MR damper.

Fig. 6. The damping force response time history for both the reference modified B-W model and RNN model with linear increment in amplitude of a periodic displacement excitation.

was performed using the MATLAB neural network toolbox, tuned for 300 training epochs. The training performance of the network is evaluated using root mean square error (RMSE) index. The RMSE index, R, is measured based on the RMSE between the target data set and output of the trained network. The R value obtained in this regression is 0.99981 which is very close to one, indicating a high accuracy obtained by the proposed model when compared with the reference modified B-W model.

4 RESULTS AND DISCUSSION In this section, several experiments are performed on the RNN model of the MR damper to compare its accuracy with the reference model. The training procedure in the previous section and the validation tests in this section are performed using the MATLAB ODE 45 (Dormand–Prince) variable time step solver, which is a smart technique to adapt the time step due to variations of simulation parameters. This feature allows complex differential equations to be solved without encountering infinity values as well as avoiding the bifurcations or chaotic response effects. The first validation test performed is based on a fixed frequency sinusoidal with linear increment in amplitude that is fed as a displacement excitation whereas the input voltage is fixed at 3 volts. Figure 6 demonstrates the output damping force generated by the reference model and the RNN model during the 5 seconds of simulation run. The force–displacement graph in Figure 7 is a testimonial of the capability of the proposed model to reproduce the hysteretic behavior of the reference model in different displacement ranges. In the next test, an input voltage with linear increases in its amplitude accompanied by a sinusoidal displace-

Fig. 7. Force–displacement graphs for both the reference and RNN models.

Fig. 8. Time history of damping force response for both models under linear incremental voltage and fixed amplitude harmonic displacement.

ment signal with constant frequency has been fed to both the reference and the RNN models. Increasing the input voltage level, an increment in output damping force of the device is expected as presented in Figure 8. The force–velocity plane for this test is also illustrated in Figure 9, which gives a good view of the


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Fig. 9. Force–velocity plane for both models under linear incremental voltage and fixed amplitude periodic displacement.

hysteretic behavior of the device in different voltages. Because the sinusoidal displacement in this experiment has fixed amplitude, the velocity perceived by the damper is a sinusoidal signal; although the velocity signal varies during the time, it has a fixed amplitude as well as displacement signal. In such case, the damping force level produced by the system will significantly depend on the input voltage. Although in many studies, it has been inferred that displacement and even acceleration can be considered as one of the main input signals for the MR dampers, our results show that the most effective input parameters for such systems are the velocity of the piston rod and the input power (voltage or current) only. Thus, including the displacement or acceleration as the system’s inputs may not be necessary. The only variable that can be fed as a control signal to the system is the input power (in the form of voltage or current). Also, because the control signal does not follow any predefined trend, this will cause the input voltage for the damper in the process to be a random variable. Hence, one major criterion for the examination of an MR damper model is to look at the performance of the model under a random input power accompanied by cyclic variations in its other inputs. This is performed in the next validation test, where a sinusoidal velocity with 0.5 cm/second and a random Gaussian voltage is fed to both the B-W and RNN models during the 30 seconds of simulation run. The force–velocity curves for B-W and RNN models are shown in Figures 10a and b, respectively, which demonstrate the hysteretic range covered by the experiment inputs and also the similarity of both models in their output wave form patterns. The output damping force generated by both models are displayed in Figure 11. It can be observed that

Fig. 10. Force–velocity plane for (a) B-W and (b) RNN models due to a cyclic velocity and random input voltage.

Fig. 11. Time history of damping force response for both the reference and RNN models under a random variation of voltage and variable frequency cyclic displacement excitation.

the RNN model can follow the output force variations due to the cyclic variations in the velocity (appearing as a cyclic pattern in the output force). It can also reconstruct the damping force even when the random input voltage has caused a chaotic effect on the output force in each velocity cyclic range (each output’s peak-to-peak interval).

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Table 3 Error indices for various evaluation tests of RNN model with respect to the B-W model

Outputs and errors Test Fixed sinusoidal displacement Sinusoidal displacement with linear amplitude increment Fixed sinusoidal displacement with linear voltage increment Sinusoidal displacement with chirp amplitude and linear frequency increment, random Gaussian voltage

RMSE

Maximum error

Absolute percentage maximum error

Absolute percentage energy error

688.1255 1, 107.3

0.048 0.023

99.98 53.39

4.76% 2.4%

2.8% 2.6%

758.248

0.0233

52.91

2.3%

3.4%

0.040

117.73

4.0%

6.7%

Maximum force (N)

Total dissipated energy (N.m)

2,099.8 2,355.8 2,270.1

2,921.7

4, 301.0

The differences of the output forces between the B-W model and the RNN model are calculated using RMSE and presented in Table 3. In this table, the maximum force and total dissipated energy are presented in the first two columns for each validation test to indicate the scales of the maximum error values. The absolute percentage maximum error (APME) is also given to show the maximum error obtained during the experiment runs. The RMSE and APME are given by the following equations: n ˆ k − yk)2 k( y (12) RMSE = n APME =

emax x100% Fmax

(13)

t = kt where n = no. of samples, yˆ k, yk are output of the RNN model and the B-W model at sample time k, respectively, and t is the sampling time in a T second cyclic motion of the device’s piston and F(t) is the force at time t. Fmax and emax are maximum force and maximum error, respectively. Absolute percentage error of the total dissipated energy (APEE) shown in Equation (16) for each test is also given in Table 3. The dissipated energy in the damper can be obtained by considering that the area under its force–displacement curve can represent the energy absorbed by the device. The area under the curve can be obtained from a discrete set of data using Equations (14) and (15) which represent the instantaneous and cyclic absorbed energy, respectively. Ek = Fk[xk − xk−1 ]

(14)

ET =

n

Fk. [(xk − xk−1 )]

(15)

ˆ T − ET E × 100% ET

(16)

i=1

APEE =

ˆ T , ET are the cyclic absorbed energy of the where E RNN model and the B-W model, respectively. The smallest RMSE is given by the third test, in which a fixed amplitude sinusoidal displacement and a linear increasing voltage are fed to the process; although the maximum RMSE is obtained from the fourth test where a variable amplitude sinusoidal signal is fed as the velocity, accompanied with a random Gaussian input voltage. This is expected as the fourth test is a more rigorous test done to find the various operating range of the model. However, in both tests the RMSE are considered as small and this shows the model obtained using the RNN method is quite accurate even under rather rigorous testing. The peak errors occur in the zero velocity conditions in which the damper’s piston is in maximum stroke in a cyclic movement and the damper’s forces are reversed due to changes in the velocity direction. It is obvious that in such positions, a small shift in velocity can cause large variations of output force. From Table 3, it can be observed that the second test gives the lowest APME showing the capability of the model to give a high traceability performance for velocity variations (caused by increase in amplitude of displacement). On the other hand, the third test with varying voltage and fixed amplitude displacement shows a lower energy dissipation error rather than the second test. This phenomenon expected as the displacement’s of the amplitude variations in the second test


will alter the area under the force–displacement plane by changing the hysteretic components’ width although in the third test the hysteretic components’ width is kept constant by a fixed magnitude of sinusoidal displacement. Because the voltage variations will only vary the amplitude for the dome shape of force–displacement curve, it can be concluded in this case that the damper stroke position affects the energy dissipation pattern of the damper more than the supplied voltage. The errors given by all tests are relatively small and thus demonstrate the accuracy of the RNN model with respect to the B-W model. 5 CONCLUSION The effectiveness of the proposed numerical method in describing the dynamic behavior of a small scale MR damper has been investigated thoroughly in this article. The model has satisfactorily predicted the response of the prototype MR damper over a wide range of operating conditions accurately. An innovative contribution made in this research for MR dampers is the introduction of multi-variable stochastic waveforms using wave modulation and random signals to excite the MR dampers for identification purposes. The major advantage of this type of system stimulation was to avoid the production of large and insufficient sets of data in the data collection process. In fact, the proposed approach permits one to intelligently trace a greater domain of working space of the device using a relatively small amount of data. The persistently exciting nature of the waveforms is able to excite the system for better model identification. Although this study is not based on experimental tests of the MR damper and only a mathematical formulation of its constitutive behavior is adopted as the reference of this study, it must be mentioned that the major computational contribution of this study is to propose a methodology for developing a model that is able to represent a real MR damper in any control realization project that uses the device. Using the model, robustness of the modeling techniques can be done by varying the parameters such as displacement amplitude, frequency, etc. which may not be possible with a real test data.

REFERENCES Adeli, H. & Jiang, X. (2006), Dynamic fuzzy wavelet neural network model for structural system identification, Journal of Structural Engineering, 132, 102. Adeli, H. & Jiang, X. (2009), Intelligent Infrastructure: Neural Networks, Wavelets, and Chaos Theory for Intelligent

231

Transportation Systems and Smart Structures, CRC Press, Boca Raton, FL. Adeli, H. & Panakkat, A. (2009), A probabilistic neural network for earthquake magnitude prediction, Neural Networks, 22(7), 1018–24. Adeli, H. & Samant, A. (2000), An adaptive conjugate gradient neural network–wavelet model for traffic incident detection, Computer-Aided Civil and Infrastructure Engineering, 15(4), 251–60. Adeli, H. & Yeh, C. (1989), Perception learning in engineering design, Computer-Aided Civil and Infrastructure Engineering, 4(4), 247–56. Ashour, O., Rogers, C. A. & Kordonsky, W. (1996), Magnetorheological fluids: materials, characterization, and devices, Journal of Intelligent Material Systems and Structures, 7(2), 123–30. Atray, V. S. & Roschke, P. N. (2004), Neuro-fuzzy control of railcar vibrations using semiactive dampers, ComputerAided Civil and Infrastructure Engineering, 19(2), 81–92. Basheer, I. A. (2000), Selection of methodology for neural network modeling of constitutive hysteresis behavior of soils, Computer-Aided Civil and Infrastructure Engineering, 15(6), 445–63. Bertotti, G. & Mayergoyz, I. D. (2006), The Science of Hysteresis: Mathematical Modeling and Applications, Academic Press, Oxford, United Kingdom. Bitaraf, M., Hurlebaus, S. & Barroso, L. R. (2012), Active and semi-active adaptive control for undamaged and damaged building structures under seismic load, Computer-Aided Civil and Infrastructure Engineering, 27(1), 48–64. Bossis, G., Khuzir, P., Lacis, S. & Volkova, O. (2003), Yield behavior of magnetorheological suspensions, Journal of Magnetism and Magnetic Materials, 258, 456–58. Bouc, R. (1967), Forced vibration of mechanical systems with hysteresis, in Proceedings of the Fourth Conference on Nonlinear Oscillation, Prague, Czechoslovakia, 315. Bouc, R. & Boussaa, D. (2002), Drifting response of hysteretic oscillators to stochastic excitation. International Journal of Non-linear Mechanics, 37, 1397–406. Carlson, J. & Weiss, K. D. (1994), A growing attraction to magnetic fluids, Machine Design, 66(15), 61–4. Chang, C. C. & Roschke, P. (1998), Neural network modeling of a magnetorheological damper, Journal of Intelligent Material Systems and Structures, 9(9), 755. Cheng, F. Y., Jiang, H. & Lou, K. (2008), Smart Structures: Innovative Systems for Seismic Response Control, CRC Press, Boca Raton, FL. Chu, S., Soong, T. T. & Reinhorn, A. M. (2005), Active, Hybrid, and Semi-Active Structural Control: A Design and Implementation Handbook, Wiley & Sons, Hoboken, NJ. Cruz, C. O. & Marques, R. C. (2012) Neuro-fuzzy cost estimation model enhanced by fast messy genetic algorithms for semiconductor hookup construction, Computer-Aided Civil and Infrastructure Engineering, 27(10), 764–81. ´ Cundumi, O. & Suarez, L. E. (2008), Numerical investigation of a variable damping semiactive device for the mitigation of the seismic response of adjacent structures, ComputerAided Civil and Infrastructure Engineering, 23(4), 291–308. Dharia, A. & Adeli, H. (2003), Neural network model for rapid forecasting of freeway link travel time, Engineering Applications of Artificial Intelligence, 16(7), 607–13. Diamatis, M., Papageorgiou, M., Kosmatopoulos, E. B. & Chamilothoris, G. (2011), Identification and adaptive control for open channel water flow systems, ComputerAided Civil and Infrastructure Engineering, 26(6), 464–80.

232

Khalid et al.

Dobson, S., Noori, M., Hou, Z., Dimentberg, M. & Baber, T. (1997), Modeling and random vibration analysis of SDOF systems with asymmetric hysteresis, International Journal of Non-Linear Mechanics, 32(4), 669–80. Du, H. & Zhang, N. (2009), Model-based fuzzy control for buildings installed with magneto-rheological dampers, Journal of Intelligent Material Systems and Structures, 20(9), 1091. Duan, Y., Ni, Y. & Ko, J. (2005), State-derivative feedback control of cable vibration using semiactive magnetorheological dampers, Computer-Aided Civil and Infrastructure Engineering, 20(6), 431–49. Dyke, S., Sain, M. & Carlson, J. (1996), Modeling and control of magnetorheological dampers for seismic response reduction, Smart Materials and Structures, 5, 565. Dyke, S., Sain, M. & Carlson, J. (1998), An experimental study of MR dampers for seismic protection, Smart Materials and Structures, 7, 693. Ghosh-Dastidar, S. & Adeli, H. (2003), Wavelet-clusteringneural network model for freeway incident detection, Computer-Aided Civil and Infrastructure Engineering, 18(5), 325–38. Goodwin, G. C. & Sin, K. S. (1984), Adaptive Filtering, Prediction and Control, Prentice Hall Inc., Englewood Cliffs, NJ. Graf, W., Freitag, S., Kaliske, M. & Sickert, J. U. (2010), Recurrent neural networks for uncertain time-dependent structural behavior, Computer-Aided Civil and Infrastructure Engineering, 25(5), 322–23. Graf, W., Freitag, S., Sickert, J. U., & Kaliske, M. (2012), Structural analysis with fuzzy data and neural networkbased material description, Computer-Aided Civil and Infrastructure Engineering, 27(9), 640–54. Guglielmino, E., Sireteanu, T., Stammers, C. W., Ghita, G. & Giuclea, M. (2008), Semi-Active Suspension Control: Improved Vehicle Ride and Road Friendliness, Springer Verlag, London, United Kingdom. Hiemenz, G. J., Choi, Y. T. & Wereley, N. M. (2003), Seismic control of civil structures utilizing semi-active MR braces, Computer-Aided Civil and Infrastructure Engineering, 18(1), 31–44. Ikhouane, F. & Rodellar, J. (2007), Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model, Wiley-Interscience, John Wiley & Sons, West Sussex, England. Jiang, X. (2003), Neuro-fuzzy logic model for freeway work zone capacity estimation, Journal of Transportation Engineering, 129, 484. Jiang, X. & Adeli, H. (2005), Dynamic wavelet neural network for nonlinear identification of highrise buildings, ComputerAided Civil and Infrastructure Engineering, 20(5), 316–30. Jiang, X. & Adeli, H. (2008a), Dynamic fuzzy wavelet neuroemulator for non-linear control of irregular building structures, International Journal for Numerical Methods in Engineering, 74(7), 1045–66. Jiang, X. & Adeli, H. (2008b), Neuro-genetic algorithm for non-linear active control of structures, International Journal for Numerical Methods in Engineering, 75(7), 770–86. Jiang, X., Mahadevan, S. & Adeli, H. (2007), Bayesian wavelet packet denoising for structural system identification, Structural Control and Health Monitoring, 14(2), 333–56.

Johnson, E. A., Christenson, R. E. & Spencer Jr., B. F. (2003), Semiactive damping of cables with sag, Computer-Aided Civil and Infrastructure Engineering, 18(2), 132–46. Kim, H. S. & Roschke, P. N. (2006), Fuzzy control of base-isolation system using multi-objective genetic algorithm, Computer-Aided Civil and Infrastructure Engineering, 21(6), 436–49. Kim, Y. (2007), Nonlinear identification and control of building structures equipped with magnetorheological dampers Ph.D. dissertation, Department of Civil Engineering, Texas A&M University, TX. Kim, Y., Hurlebaus, S. & Langari, R. (2010), Model-based multi-input, multi-output supervisory semi-active nonlinear fuzzy controller, Computer-Aided Civil and Infrastructure Engineering, 25(5), 387–93. Kwok, N., Ha, Q., Nguyen, T., Li, J. & Samali, B. (2006), A novel hysteretic model for magnetorheological fluid dampers and parameter identification using particle swarm optimization, Sensors and Actuators A: Physical, 132(2), 441–51. Lam, H. F., Yuen, K. V. & Beck, J. L. (2006), Structural health monitoring via measured ritz vectors utilizing artificial neural networks, Computer-Aided Civil and Infrastructure Engineering, 21(4), 232–41. Macki, J. W., Nistri, P. & Zecca, P. (1993), Mathematical models for hysteresis, SIAM Review, 35(1), 94–123. Mayergoyz, I. D. (2003), Mathematical Models of Hysteresis and Their Applications, Elsevier Science (Academic Press), New York. Mills, P. M., Zomaya, A. Y. & Tade, M. O. (1995), NeuroAdaptive Process Control: A Practical Approach, John Wiley & Sons, West Sussex, England. Panakkat, A. & Adeli, H. (2009), Recurrent neural network for approximate earthquake time and location prediction using multiple seismicity indicators, Computer-Aided Civil and Infrastructure Engineering, 24(4), 280–92. Park, H. S. & Adeli, H. (1995), A neural dynamics model for structural optimization–application to plastic design of structures, Computers & Structures, 57(3), 391–99. Park, H. S. & Adeli, H. (1997), Distributed neural dynamics algorithms for optimization of large steel structures, Journal of Structural Engineering, 123, 880. Peschel, M. & Roschke, P. N. (2001), Neuro-fuzzy model of a large magnetorheological damper, in Proceedings Texas Section ASCE, Spring Meeting, San Antonio, TX, 241–46. Ruano, A. E. (2005), Intelligent Control Systems Using Computational Intelligence Techniques, The Institution of Engineering and Technology, London, United Kingdom. Schurter, K. C. & Roschke, P. N. (2000), Fuzzy modeling of a magnetorheological damper using ANFIS, 9th IEEE International Conference on Fuzzy Systems, 1, 122–27. Sodeyama, H., Sunakoda, K., Fujitani, H., Soda, S., Iwata, N. & Hata, K. (2003), Dynamic tests and simulation of magneto–rheological dampers, Computer-Aided Civil and Infrastructure Engineering, 18(1), 45–57. Spencer, B., Dyke, S., Sain, M. & Carlson, J. D. (1997), Phenomenological model for magnetorheological dampers, Journal of Engineering Mechanics, 123(3), 230–38. Sunakoda, K. (2004), Experimental behavior and neuro-fuzzy modeling of 30-ton magnetorheological damper, KSCE Journal of Civil Engineering, 8(2), 213–19. Visintin, A. (1994), Differential Models of Hysteresis, Springer Verlag, Berlin, Germany.


Wan, F., Wang, L.-X., Zhu, H.-Y. & Sun, Y.-X. (2001), Generating persistently exciting inputs for nonlinear dynamic system identification using fuzzy models, 10th IEEE International Conference on Fuzzy Systems, 1, 505–08. Wang, D. & Liao, W. (2005), Modeling and control of magnetorheological fluid dampers using neural networks, Smart Materials and Structures, 14, 111. Wang, L. X. (1994), Adaptive Fuzzy Systems and Control Design and Stability Analysis, Prentice Hall, Englewood Cliffs, NJ.

233

Wen, Y. K. (1976), Method for random vibration of hysteretic systems, Journal of the Engineering Mechanics Division, 102(2), 249–63. Yuen, K. V., Shi, Y., Beck, J. L. & Lam, H. F. (2007), Structural protection using MR dampers with clipped robust reliability-based control, Structural and Multidisciplinary Optimization, 34(5), 431–43. Zapateiro De La Hoz, M. F. (2009), Semiactive control strategies for vibration mitigation in adaptronic structures equipped with magnetorheological dampers. Ph.D. thesis, University of Girona, Spain.