A novel hysteretic model for magnetorheological fluid dampers and parameter identification using particle swarm optimization

A Novel Hysteretic Model for Magnetorheological Fluid Dampers and Parameter Identification Using Particle Swarm Optimization N M Kwok ∗ , Q P Ha, T H Nguyen, J Li and B Samali Faculty of Engineering, University of Technology, Sydney Broadway, NSW 2007, Australia Abstract Nonlinear hysteresis is a complicated phenomenon associated with magnetorheological (MR) fluid dampers A new model for MR dampers is proposed in this paper For this, computationally-tractable algebraic expressions are suggested here in contrast to the commonly-used Bouc-Wen model, which involves internal dynamics represented by a nonlinear differential equation In addition, the model parameters can be explicitly related to the hysteretic phenomenon To identify the model parameters, a particle swarm optimization (PSO) algorithm is employed using experimental force-velocity data obtained from various operating conditions In our algorithm, it is possible to relax the need for a priori knowledge on the parameters and to reduce the algorithmic complexity Here, the PSO algorithm is enhanced by introducing a termination criterion, based on the statistical hypothesis testing to guarantee a user-specified confidence level in stopping the algorithm Parameter identification results are included to demonstrate the accuracy of the model and the effectiveness of the identification process Key words: magnetorheological damper, modelling, particle swarm optimization ∗ Corresponding author Email address: ngai.kwok@eng.uts.edu.au (N M Kwok) Preprint submitted to Sensors and Actuators A, physical March 2006 Introduction Vibration suppression may be considered as a key component in the performance of civil engineering and mechanical structures for the safety and comfort of their occupants In [1], a survey was conducted in the context of building control where the magnetorheological (MR) damper as a semi-active device was introduced The MR damper may be constructed in the cylinder-piston or pin-rotor form [2], which makes it widely applicable in various domains, e.g., vehicle suspensions [3] The actuation of MR dampers is governed by tiny magnetizable particles which are immersed in a carrier fluid and upon the application of an external magnetic field aligned in chain-like structures, see [4] The alignment of particles changes the yield stress of the fluid and hence produces a controllable damping force The MR damper is an attractive candidate in vibration suppression applications in which only a small amount of energizing power is required and the fluid characteristic is reversible in the range of milliseconds The damper also features a fail-safe mode, acting as a conventional damper, and in case of hazardous situations as encountered in earthquakes where power supply may be interrupted Although the MR damper is promising in control applications, its major drawback lies in the non-linear and hysteretic force-velocity response Furthermore, the design of a controller generally requires a model of the actuator which may be challenging in the case of employing the MR damper The modelling of the hysteresis had been studied in [5] and [6] including the Bingham visco-plastic model, the Bouc-Wen model, the modified Bouc-Wen model and many others These models range from simple dry-friction to complicated differential equation representations However, it is noted of a trade-off between the model accuracy and its complexity From the control engineering point of view, nonlinear differential equation based models may affect robustness of the control system and hamper the feasible controller realization as in the case of MR dampers employed in the reduction of seismic response in buildings [7] On the other end of the spectrum, polynomial based modelling of the MR damper was attempted in [8] with a reduction in model complexity A hyperbolic function based curve-fitting model was proposed in [9] with satisfactory results A black-box damper model was also applied in [10] as an alternative Soft-computing techniques, fuzzy inference systems and neural networks were also applied in modelling a MR damper; see [11] and [12], which represent another paradigm for a suitable approach towards an efficient model Evolutionary computation methods, e.g., genetic algorithms [13], [14], have also been widely applied in modelling and parameter identification applications and many others In [15], the genetic algorithm (GA) was employed to identify a mechatronic system of unknown structure However, in addition to the implementation complexity, the GA may found difficulties in convergence to optimal parameters unless elitism [16] is explicitly incorporated in the algorithm An identification procedure following this approach was reported in [17] Moreover, a priori knowledge on the ranges of solutions may be required to accelerate the convergence rate A recently developed optimization technique, the particle swarm optimization (PSO), has been recognized as a promising candidate in applications to model parameter identification when the identification is cast as an optimization problem The PSO is based on the multi-agent or population based philosophy (the particles) which mimics the social interaction in bird flocks or schools of fish, [18] By incorporating the search experiences of individual agents, the PSO is effective in exploring the solution space in a relatively smaller number of iterations, see [19] In emulating bird flocks, particles are assigned with velocities that lead their flight across the solution space The best solutions found by an individual particle and by the whole population are memorized and used in guiding further search flights The best solution is reported at the satisfaction of some termination criteria Other applications of the PSO can be found in the design of PID controllers [20] and electro-magnetics [21] The PSO convergence characteristic was analyzed in [22], where algorithm control settings were also proposed In this work, a novel model for the MR damper will be proposed This model uses a hyperbolic tangent function to represent the hysteretic loop together with components obtained from conventional viscous damping and spring stiffness This approach, as an attractive feature, maintains a relationship between the damper parameters and physical force-velocity hysteretic phenomena and reduces the complexity from using a differential equation to describe the hysteresis The PSO is then applied to identify the model parameters using experimental force-velocity data obtained from a commercial MR damper Here, to enhance the identification process, a statistical hypothesis testing procedure is adopted to determine the termination of the PSO This procedure is able to guarantee a user defined level of confidence on the quality of the identified parameters The rest of the paper is organized as follows In Section 2, the MR damper is introduced and the commonly-used models are reviewed The new model is suggested in Section with a discussion of the model parameters The PSO, as applied in identifying the proposed model parameters, is presented in Section and its advantages are highlighted Control settings for the algorithms are determined and a performance-enhanced criterion is also proposed Results of parameter identification are given in Section together with some discussion Finally, a conclusion is drawn in Section Magnetorheological Damper 2.1 Principle of Operation The magnetorheological damper may be viewed as a conventional hydraulic damper except that the contained fluid is allowed to change its yield stress upon the application of an external magnetic field The structure of the damper is sketched in Fig Fig MR damper structure Tiny magnetizable particles, e.g., carbonyl iron, are carried in non-magnetic fluids such as silicon oil and are housed within a cylinder In most applications, the damping force is generated in the flow-mode [23] The yield of the MR fluid changes inversely to the temperature and a reduction in the damper force results with an increase in temperature However, an appropriately designed current controller can be applied to compensate for the change in the damper force This controller will also be employed for counteracting the long-term stability problem of the MR fluid The MR damper used in this work is a RD1005-3 model manufactured by the LORD Corporation The damper has a compressed length of 155mm, weighs 800g, accepts a maximum input current of 2A at 12V DC and the response time is less than 25msec Interested readers are referred to the product information provided by the manufacturer [24] 2.2 Damper Characteristics In order to gain an insight into the MR damper characteristics, experimental data are obtained from our laboratory with the test gear shown in Fig Force Sensor MR Damper Drive Fig MR damper test gear A sinusoidal excitation of small magnitude (e.g., 4mm-12mm) and at low frequencies (e.g., 1Hz-2Hz) is applied from a hydraulic drive to the damper as the damper displacement The damper forces generated, under the application of a set of magnetizing currents (0-2A) are measured by a force sensor (load cell) mounted on the upper end of the damper The displacement and the damper force readings are recorded for parameter identification A typical damper force plot is depicted in Fig The force-displacement non-linearity is very noticeable in Fig 3(a), and the hysteresis is observed in Fig 3(b) For dampers operating linearly the generated force is proportional to the viscous damping and spring stiffness coefficients respectively This results in an elliptical plot for the force/displacement relationship and an inclination of the ellipse in the force/velocity response However, due to non-linearity, there are discontinuities occurring at the extremes of the damper piston stroke travel limits of ±8mm, in our experiments The discontinuity also appears in a lagging effect of the force with respect to the velocity, within the ±20mm/s range, and produces the hysteretic phenomenon as shown Our experiments were conducted at room temperature, around 25◦ C The study on the temperature effect, which has certain influences on the MR fluid properties [23], is out of the scope of this research 1000 1000 2.00A 2.00A 1.00A 1.00A 0.50A 0.25A 0.25A Force(N) Force(N) 0.50A 0.00A 0.00A −500 −500 −1000 −1000 −1500 −10 −8 −6 −4 −2 0.75A 500 0.75A 500 −1500 10 −60 −40 −20 20 40 60 Velocity(mm/s) Displacement(mm) (a) (b) Fig Characteristics of damper force vs supply current: (a) non-linearity in force-displacement; (b) hysteresis in force-velocity 2.3 Damper Models Various models had been proposed to represent the hysteretic behaviour of the MR damper, [5] and [6] The following models are among the most commonly employed in previous work 2.3.1 Bingham Model The stress-strain visco-plastic behaviour is used in the Bingham model The model contains a dead-zone or a discontinuous jump in the damper force/velocity response The damper force is expressed as f = fc sign(x) ˙ + c0 x˙ + f0 , (1) where f is the damper force, fc is the magnitude of hysteresis, sign(·) is the signum function, x˙ is the velocity, x is the displacement, c0 is the viscous coefficient and f0 is an offset of the damper force Although the model is simple, the hysteresis cannot be adequately described, e.g., roll-off effects Therefore, this model is only employed in situations when there is a significant need for a simple model 2.3.2 Bouc-Wen Model This model contains components from a viscous damper, spring and a hysteretic component The model can be described by the force equation and the associated hysteretic variable, given by f = cx˙ + kx + αz + f0 z˙ = δ x˙ − β x|z| ˙ n − γz|x||z| ˙ n−1 , (2) (3) where α, δ, β, γ, n are the model parameters and z is the hysteretic variable Notice that when α = 0, the model represents a conventional damper The Bouc-Wen model is most commonly-used to describe the MR damper hysteretic response The number of parameters is less than that for the modified Bouc-Wen model (see below) The modelling accuracy is practically acceptable resulting from a trade-off between accuracy and complexity However, due to the incorporation of internal dynamics with respect to the damper state variable z, undesirable singularities may occur during the identification process 2.3.3 Modified Bouc-Wen Model In the modified Bouc-Wen model, additional parameters are introduced in order to obtain a more accurate model It is given as f = c0 (x˙ − y) ˙ + k0 (x − y) + k1 x + αz + f0 y˙ = (c0 + d1 )−1 (c0 x˙ + k0 (x − y) + αz) n z˙ = δ(x˙ − y) ˙ − β(x˙ − y)|z| ˙ − γz|x˙ − y||z| ˙ n−1 , (4) (5) (6) where y is an internal dynamical variable, d1 and k1 are additional coefficients of the added dashpot and spring in the model It has been shown in [5] that the modified Bouc-Wen model improves the modelling accuracy However, the model complexity is unavoidably increased with an extended number of model parameters which may impose difficulties in their identification Therefore, this model is only used in applications where an accurate model is required Proposed MR Damper Model A simple model is proposed here to model the hysteretic force-velocity characteristic of the MR damper A component-wise additive strategy is employed which contains the viscous damping (dashpot), spring stiffness and a hysteretic component, Fig illustrates the conceptual configuration Fig Hysteresis model - component-wise additive approach 3.1 Mathematical Model In terms of mathematical expressions, the model makes use of a hyperbolic tangent function to represent the hysteresis and linear functions to represent the viscous and stiffness The model is given by f = cx˙ + kx + αz + f0 z = tanh(β x˙ + δsign(x)), (7) (8) where c and k are the viscous and stiffness coefficients, α is a scale factor of the hysteresis, z is the hysteretic variable given by the hyperbolic tangent function and f0 is the damper force offset Note that the model contains only a simple hyperbolic tangent function and is computationally efficient in the context of parameter identification and subsequent inclusion in controller design and implementation A description and an analysis of the parameters will be given in the next subsection 3.2 Relationship Between Parameters and Hysteresis The components building up the hysteresis are depicted in Fig which illustrates the effects of the parameters on the damper force-velocity response The viscosity cx˙ generates an inclined line that represents the post-yield (at the two ends of the hysteresis) relationship between the velocity and the damper force Large coefficient c gives a steep inclination The stiffness k, (the horizontal ellipse formed from the product kx) is responsible for the opening found from the vicinity of zero velocity A large value of k corresponds to the opening of the ends Parameters c and k contribute to the representation of a conventional damper without hysteresis 400 300 β 200 c Force(N) 100 k −100 δ f −200 −300 −400 Final Hysteresis −500 −600 −30 −20 −10 10 20 30 Velocity(mm/s) Fig Hysteresis parameters The basic hysteretic loop, which is the smaller one shown in Fig 5, is determined by β This coefficient is the scale factor of the damper velocity defining the hysteretic slope Thus a steep slope results from a large value of β The scale factor δ and the sign of the displacement determine the width of the hysteresis through the term δsign(x), a wide hysteresis corresponds to a large value of δ The overall hysteresis (the larger hysteretic loop shown in Fig 5) is scaled by the factor α determining the height of the hysteresis The overall hysteretic loop is finally shifted by the offset f0 After the development of the simple model, we proceed to identify the model parameters and the approach adopted will be detailed in the following sections Enhanced Particle Swarm Optimization The particle swarm optimization (PSO) is inspired by the social interaction and individual experience [18], observed through human society development and natural behaviours of bird flocks and fish schools This technique is a population based and controlled heuristic search and has been applied in a wide domain in function optimizations, [19] In the following, the PSO is compared with its counterpart, the genetic algorithm (GA), to highlight its potential in reduction of the computational complexity The performance of the algorithm is further enhanced with a proposed termination criterion 4.1 Comparison of PSO with GA The GA, developed before the PSO, is also a population based search algorithm inspired by natural evolution It is widely applicable, for example, in system identification, control, planning and scheduling and many others, see [13] and the references therein A basic GA can be described by the pseudo code shown in the following Initialize random population While not terminate Evaluate fitness Do selection Do crossover and mutation Report best solution if terminate The ability to obtain a near-optimal solution is guaranteed by the Schemata theory [14], the quality of the solution is a trade-off between accuracy and computational load in terms of the number of iterations needed The operation of the algorithm can be viewed as a concentration of search areas, i.e., exploitation, through the selection and crossover operator The search ability is enhanced through the use of the mutation operator with regard in exploration However, the exploration process is generally slow and this may increase the search time span when the initial population does not cover the solution However, in practice, a priori knowledge on the solution that can be used to guide the initialization of the population may not always be available Furthermore, the crossover operator may destroy useful solutions Hence, an elitism strategy is usually implemented [16] which may also increase the computational load In PSO implementations, a particle represents a potential solution The values of the particles are continuously adjusted by emulating the particles as bird flights That is, each particle is assigned a velocity to update its value as a new position The operation of the PSO is described by the following equations [19] assuming a unity sampling time vt+1 = ωvt + c1 (xg − xt ) + c2 (xp − xt ) xt+1 = xt + vt+1 , (9) (10) where vk is the present particle velocity, ω is the velocity scale factor, c1 and c2 are uniform random numbers c1 ∈ [0, c1,max ] and c2 ∈ [0, c2,max ], xg is the group best (global) solution found up to the present iteration, xp is the personal best solution found by individual particles from their search history through time index t The pseudo code for the PSO is given below Initialize random particles While not terminate Evaluate fitness Find group- and personal-best Update velocity and particle position Report group-best if terminate 10 A comparison on the computational efficiency between GA and PSO may reveal that the later is more attractive Within a single iteration in the two algorithms, the evaluation of the fitness of each potential solution is mandatory in both algorithms The selection and crossover operator in GA are two-pass operations while a small number of mutation operations are conducted On the other hand, the finding of group and personal best are single-pass operations The updates of velocity and position are simple additions Furthermore, the generation of random numbers, crossover/mutation operation in GA and scaling by c1 and c2 in PSO are common to both algorithms with similar complexities The saving in PSO computation is obtained from its simplified calculations for the group- and personal-best particle In addition, PSO inherently maintains the group-best without an explicit elitism implementation 4.2 PSO Control Settings The control settings of the PSO can be obtained by conducting an analysis using control system theories, see [22] The confidence on the optimization result may be derived from an indication of the particles being concentrated in the vicinity of the group-best particle xg The philosophy adopted is that of the convergence of the potential solutions to the optimal Now, denote the position error of a particle as εt+1 = xg − xt (11) Following Eq and 10, the particle position error and velocity can be written in the state-space form as     εt+1   − c1 − c2  = vt+1 c1 + c2   −ω   εt  ω  vt     +  (xg − xp ), (12) −1 or zt+1 = Azt + But , (13) where zt+1 = [εt+1 , vt+1 ]T , ut = xg − xp , and A, B are self-explanatory It becomes clear that the requirement for convergence implies εt → and vt → as time t → ∞ When the best solution is found xg becomes a constant and xp will tend to xg if the system is stable The stability of a discrete control system can be ascertained by constraining the magnitude of the eigenvalues 11 λ1,2 of the system matrix A ∈ R2×2 to be less than unity, that is λ1,2 ∈ {λ|λ2 − (1 + ω − c1 − c2 )λ + ω = 0}, |λ1,2 | < (14) By choosing the maximal random variables c1 and c2 to be c1,max = and c2,max = and take the expectation values from a uniform distribution, the coefficients become c1 = and c2 = This case corresponds to a total feedback of the discrepancy of the particle positions from the desired solution at xg Writing out the eigenvalues, we have λ1,2 = √ ω − ± − 6ω + ω (15) After some manipulations, it can be shown that ω < with c1,max = c2,max = will guarantee stability for the system, hence particle will converge to xg Here ω = 0.65 is used in the MR damper model identification for a moderate rate of convergence 4.3 Enhanced Termination Criteria The PSO is basically a recursive algorithm that iterates until some termination criteria is met In common practice, the termination criteria may be defined as the expiry of a certain number of iterations An alternative strategy usually employed is to check if the improvement on the best fitness diminishes or not However, it is desirable that the termination of the iteration will be aligned to a user specified degree of confidence Here, the termination of the PSO is cast as a statistical hypothesis test between a null hypothesis and an alternative The action according to the null hypothesis is to continue the iteration, while the alternative action is to terminate but with a specific bound on the decision error The hypothesis can be stated as H0 : there will be improvements in the fitness H1 : there will be no improvement in the fitness (16) (17) From the structure of the PSO, it is noticed that the algorithm explicitly maintains the group-best xg with the associated minimum fitness (e.g., where a minimization problem is considered) For a minimization problem as considered here, the fitness, f it, corresponding to xg will not increase as the 12 algorithm evolves That is, f itt−1 ≥ f itt (18) Alternatively, the improvement in fitness is a positive variable described as ∆f it ≡ f itt−1 − f itt ≥ (19) After a certain number of iterations, statistical data can be collected Moreover, when the best solution is found, the fitness improvement quantity becomes zero It is concluded that this quantity can be approximated by an exponential distribution as p(∆f it) ∼ ∆f it exp(− ), λ λ (20) where the symbol (∼) stands for sampling from a distribution and λ−1 is the mean of ∆f it The corresponding probability for ∆f it to fall within some threshold γ is P (∆f it ≤ γ) ≡ P (γ) = − exp(−γ), (21) where γ is to be determined in order to terminate the PSO algorithm If the next fitness improvement ∆f it falls within the threshold γ then one can ascertain a confidence of − P (γ) Fig illustrates this concept Here, we propose a threshold of γ = 0.1λ−1 , the associated error in making a decision to terminate the PSO is then: P (∆f it ≤ γ) = − exp(−0.1) = 0.095 (22) The strategy adopted here is to fix γ and accept H0 until the mean of ∆f it falls below the threshold γ = 0.1λ−1 Consequently, this can guarantee a specific level of confidence (say − 0.095 = 0.905 as indicated above) in terminating the PSO algorithm Identification Results Experimental data, including the damper displacement, velocity and the generated damper force were collected under a wide range of operating conditions 13 0.5 0.45 0.4 0.35 −1 P(0.1λ )=0.095 p(∆fit) 0.3 0.25 0.2 0.15 0.1 0.05 −1 λ =2 ∆fit 10 Fig PSO termination determined from an exponential distribution as described in Section 2.2 The current supplied to the damper was varied from 0A to 2A, the driving frequencies were set at 1Hz and 2Hz while the displacement ranged from 4mm to 12mm respectively The test conditions are summarized in Table below Note that there are six combinations of frequency and displacement, and for each combination there are six current settings Table Test conditions Current(A) 0.00 0.25 0.50 0.75 1.00 2.00 Frequency(Hz) 1 2 Displacement(mm) 12 The data set were stored in corresponding computer files and the damper model parameters are identified off-line using the proposed PSO algorithm Upon the availability of a data set with specified current, displacement and frequency values; the model proposed in Section (Eq 7) is used to simulate the hysteretic responses Each hysteretic loop is determined by a set of model parameters encoded as particles in the PSO algorithm This gives an array of N -rows and M -columns, where N is the number of particles and M is the number of parameters to be identified, i.e., c, k, α, f0 , β and δ In this work, the number of particles used is set at 50 while the control coefficients are set as described in Section 4.2 (c1 = c2 = 2, ω = 0.65) and the termination criteria is determined according to the development given in Section 4.3 The fitness is evaluated as the root-mean-square error between the experimen14 tal and simulated damper force, that is, n i (F i − Fsim )2 , n i=1 exp f itt = (23) i is the damper force from the where n is the number of data points, Fexp i experiment and Fsim is the simulated force from the proposed model The effectiveness of the parameter identification process will be assessed in terms of the shape of the reconstructed hysteresis, reconstruction error and computational efficiency 5.1 Reconstructed Hysteresis Hysteresis loops are reconstructed or simulated from models using the identified parameter sets for the proposed and Bouc-Wen models Fig shows the reconstructed hysteresis from the proposed model while Fig shows the reconstructed hysteresis for the Bouc-Wen model, both in different test conditions The damper forces obtained from experiments are plotted in dots while the model predictions are plotted in solid lines Freq:1.00Hz Disp:8.00mm 1000 Freq:2.00Hz Disp:4.00mm 1000 2.00A 2.00A 1.00A 500 1.00A 0.75A 500 0.75A 0.50A 0.50A 0.25A 0.25A Force(N) Force(N) 0.00A −500 −500 −1000 −1500 −0.06 0.00A −1000 −0.04 −0.02 Velocity(m/s) 0.02 0.04 0.06 (a) −1500 −0.05 −0.04 −0.03 −0.02 −0.01 0.01 Velocity(m/s) 0.02 0.03 0.04 0.05 (b) Fig Reconstructed hysteresis from proposed model: (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm The smoothness of the reconstructed hysteresis from the proposed model is obtained from the hyperbolic tangent function On the other hand, the BoucWen model is capable to produce sharp curves because of its representation of hysteresis by a non-linear differential equation However, this may lead to the problem of over-fitting where measurement imperfections are being captured by the model 15 Freq:1.00Hz Disp:8.00mm 1000 Freq:2.00Hz Disp:4.00mm 2.00A 1000 2.00A 1.00A 500 1.00A 500 0.75A 0.75A 0.50A 0.50A 0.25A 0.25A 0.00A Force(N) Force(N) 0.00A −500 −500 −1000 −1000 −1500 −0.06 −0.04 −0.02 Velocity(m/s) 0.02 0.04 −1500 −0.05 0.06 −0.04 −0.03 −0.02 (a) −0.01 0.01 Velocity(m/s) 0.02 0.03 0.04 0.05 (b) Fig Reconstructed hysteresis from Bouc-Wen model: (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm 5.2 Identification Errors 120 120 100 100 80 80 RMS Error RMS Error The errors between the damper forces obtained from the experimental data and simulated force from the models using the identified parameters are show in Fig Due to the higher degree of non-linearity in the Bouc-Wen model, a larger root-mean-square error is observed from the reconstructed or simulated force from the model On the other hand, the errors found from the proposed model are generally less than that from the Bouc-Wen model 60 60 40 40 20 20 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Current(A) 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Current(A) (a) (b) Fig Parameter identification errors: (a) from proposed model; (b) form Bouc-Wen model Legends ◦: 1Hz 4mm, : 1Hz 8mm, ♦: 1Hz 12mm, : 2Hz 4mm, ✁: 2Hz 6mm and ✄: 2Hz 8mm 5.3 Computation Efficiency The efficiency of the identification process is affected by the complexity of the model and the number of parameters to be identified The proposed model 16 35 35 30 30 Generations Generations is simpler and the number of parameters is smaller Hence, the identification efficiency out-performs that from the Bouc-Wen model as show in Fig 10 (for legends, see Fig 9) The identification of the proposed model terminates around 15 generations, while for the Bouc-Wen model it requires over 30 generations before the same PSO algorithm terminates Furthermore, the spread in the number of generations before termination is smaller in the proposed model This feature makes the proposed model more efficient and predictable in the paradigm of computation efficiency 25 20 15 10 25 20 15 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 10 0.2 0.4 0.6 0.8 Current(A) Current(A) (a) (b) 1.2 1.4 1.6 1.8 Fig 10 Parameter identification efficiency: (a) from proposed model; (b) from Bouc-Wen model 5.4 Generalized Parameters The identified parameters are grouped (e.g., the viscous parameter - c) according to their experiment settings and are plotted against the supplied current as shown in Fig 11 (for legends, see Fig 9) The parameter groups are then averaged and a polynomial is used to fit the averaged values This results of this operation (shown by dotted lines in the figure) give the following set of expressions describing the parameters as functions of the supply current c = 1929i + 1232 k = −1700i + 5100 α = −244i2 + 918i + 32 β = 100 δ = 0.30i + 0.58 f0 = −18i − 257, (24) (25) (26) (27) (28) (29) where i is the current supplied to the MR damper Most of the parameters c, k, δ and f0 can be approximated using a 1st-order 17 polynomial and the relationships between the parameters and the current become linear An exception is observed from the scaling parameter α where a 2nd-order polynomial is needed to represent the relationship to the supply current It is observed that, in particular, parameter β is approximately a constant against the supply current values Consider the change of parameters with regard to the limits of allowed range of current supplied to the damper, i.e., from 0A to 2A The minimum parameter ratio is obtained from the offset f0 at : 1.5 to the maximum ratio of : obtained for the viscous damping parameter c Furthermore, these parameters change linearly with the supply current (approximated by a 1st-order polynomial) This characteristic makes the proposed model very suitable in implementing controllers for vibration reduction in buildings The set of polynomial fitted parameters are used to reconstruct the hysteretic responses and compared to the experimental data shown in Fig 12 The results indicate that the matching between the reconstructed hysteresis and experimental data is practically acceptable although the matching contains some discrepancy It is suspected that the collected experimental data are not depicting the physical hysteresis accurately due to measurement errors For example, when the displacement is set at as small as 4mm, it becomes practically difficult to precisely mount the test gears or there are dead-zone or back-slash found in the drive mechanism Another problem may be encountered is the over-fitting phenomena while the PSO algorithm is directed to fit a noise corrupted or distorted hysteresis, thus giving rise to the discrepancy Conclusion This paper has presented a new model for a magnetorheological damper to represent the hysteretic relationship between the damping force and the velocity Complexity arising from a larger number of model parameters, when using the Bouc-Wen model, is removed The model parameters can be explicitly related to the hysteretic phenomenon while still maintain physical interpretations for viscous damping and spring stiffness A performance-enhanced technique based on particle swarm optimization is proposed for identifying the model parameters A statistical hypothesis testing procedure is adopted here to terminate the optimization process that guaranteed the identification quality Experiment data from a commercial MR damper is used for model validation Results obtained by the new model have shown highly satisfactory coincidence with the experimental data, and also the effectiveness of the proposed identification technique 18 1929.4424 1231.5238 2.5 −1663.2508 x 10 5133.0963 15000 Param − c Param − k 10000 1.5 5000 0.5 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 0.2 0.4 0.6 (a) −244.2819 0.8 1.2 1.4 1.6 1.8 1.2 1.4 1.6 1.8 Current(A) Current(A) (b) 917.6953 98.666 31.86744 200 1000 900 180 800 160 700 Param − β Param − α 140 600 500 400 120 100 300 80 200 60 100 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 40 0.2 0.4 0.6 0.8 (c) 0.30404 Current(A) Current(A) (d) 0.58375 −18.21205 1.6 −265.5034 −220 1.4 −240 −260 Param − f Param − δ 1.2 0.8 −280 −300 0.6 −320 0.4 −340 0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 −360 Current(A) 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Current(A) (e) (f) Fig 11 Parameter identification results (−) and polynomial fitted coefficients (· · ·): (a) parameter c; (b) parameter k; (c) parameter α; (d) parameter β; (e) parameter δ and (f) parameter f0 19 Freq:1.00Hz Disp:8.00mm Freq:2.00Hz Disp:4.00mm 1000 1000 500 2.00A 2.00A 1.00A 1.00A 500 0.75A 0.75A 0.50A 0.50A Force(N) Force(N) 0.25A 0.00A 0.00A −500 −500 −1000 −1000 −1500 −60 −40 −20 20 40 −1500 −60 60 0.25A −40 −20 Velocity(mm/s) Velocity(mm/s) (a) (b) 20 40 60 Fig 12 Results of parameter identification: experimental data (· · ·), reconstructed hysteresis from polynomial fitted parameters (−): (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm Acknowledgement This work is supported by Australian Research Council (ARC) Discovery Project Grant DP0559405, by the UTS Centre for Built Infrastructure Research and by the Centre of Excellence programme, funded by the ARC and the New South Wales State Government References [1] B F Spencer Jr and M K Sain, ”Controlling buildings: a new frontier in feedback,” IEEE Control Systems Magazine, Vol 17, No 6, pp 19-35, Dec 1997 [2] T Tse and C C Chang, ”Shear-mode rotary magnetorheological damper for small-scale structural control experiments,” J of Structural Engineering, Vol 130, No 6, pp 904-911, 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convergence in a multidimensional complex space,” IEEE Trans on Evolutionary Computation, Vol 6, No 1, pp 58-73, Feb 2002 [20] Z L Gaing, ”A particle swarm optimization approach for optimum design of PID controller in AVR system,” IEEE Trans on Energy Conversion, vol 19, No 2, pp 384-391, Jun 2004 21 [21] G Ciuprina, D Ioan and I Munteanu, ”Use of intelligent-particle swarm optimization in electromagnetics,” IEEE Trans on Magentic, Vol 38, No 2, pp 1037-1040, Mar 2002 [22] I C Trelea, ”The particle swarm optimization algorithm: convergence analysis and parameter selection,” Information Processing Letters, Vol 85, pp 317-325, 2003 [23] L Zipser, L Richter and U Lange, ”Magnetorheological fluids for actuators,” Sensors and Actuators A: Physical, Vol 92, pp 318-325, 2001 [24] ”Product Bulletin and MR Damper Assembly (RD-1005-3),” LORD Corporation, www.lord.com 22 List of Figure Captions Fig MR damper structure Fig MR damper test gear Fig Characteristics of damper force vs supply current: (a) non-linearity in force-displacement; (b) hysteresis in force-velocity Fig Hysteresis model – component-wise additive approach Fig Hysteresis parameters Fig PSO termination determined from an exponential distribution Fig Reconstructed hysteresis from proposed model: (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm Fig Reconstructed hysteresis from Bouc-Wen model: (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm Fig Parameter identification errors: (a) from proposed model; (b) form BoucWen model Fig 10 Parameter identification efficiency: (a) from proposed model; (b) from Bouc-Wen model Fig 11 Parameter identification results (-) and polynomial fitted coefficients ( ): (a) parameter c ; (b) parameter k ; (c) parameter α ; (d) parameter β ; (e) parameter δ ; and (f) parameter fo Fig 12 Results of parameter identification: experimental data ( ), reconstructed hysteresis from polynomial fitted parameters (-): (a) frequency=1Hz, displacement=8mm; (b) frequency=2Hz, displacement=4mm Paper SNA-D-05-00719 - Authors’ biography N.M Kwok received the B.Sc degree in Computer Science from the University of East Asia, Macau, the M.Phil degree in Control Engineering from The Hong Kong Polytechnic University, Hong Kong and the PhD degree in Mobile Robotics from the University of Technology, Sydney, Australia in 1993, 1997 and 2006 respectively He is currently a Senior Research Assistant at the University of Technology, Sydney, Australia His research interests include evolutionary computing, robust control and mobile robotics Q.P Ha received the B.E degree in Electrical Engineering from Ho Chi Minh City University of Technology, Vietnam, the Ph.D degree in Engineering Science from Moscow Power Engineering Institute, Russia, and the Ph.D degree in Electrical Engineering from the University of Tasmania, Australia, in 1983, 1992, and 1997, respectively He is currently an Associate Professor at the University of Technology, Sydney, Australia His research interests include robust control and estimation, robotics, and artificial intelligence applications T.H Nguyen received the B.E degree in Electrical - Electronic Engineering from Ho Chi Minh University of Technical Education, Vietnam, the Masters degree in Telecommunication - Electronic Engineering from Ho Chi Minh City University of Technology, Vietnam, in 1995 and 2001 respectively He is currently a Ph.D student at the University of Technology, Sydney, Australia J Li received his PhD in Mechanical Engineering of University of Dublin, Ireland in 1993 He is currently a senior lecturer in Faculty of Engineering University of Technology Sydney His research interests include structural dynamics, smart materials and smart structures and structural health monitoring and damage detections B Samali is the current Professor and Head of Infrastructure and the Environment disciplines at the Faculty of Engineering at the University of Technology, Sydney and has a personal chair in Structural Engineering at UTS He is also the Director of Centre for Built Infrastructure Research at UTS He received his Bachelor of Science in Civil Engineering with honours in 1978, Master of Science in Structural Engineering in 1980 and Doctor of Science in Structural Dynamics in 1984, all from the George Washington University in Washington DC (USA) Professor Samali has published over 250 technical papers in engineering journals and conference proceedings His main research interests lie in the general area of structural dynamics including wind and earthquake engineering with special emphasis on structural control, dynamic measurement and analysis of buildings and bridges, and use of smart materials in engineering applications  ... coefficients ( ): (a) parameter c ; (b) parameter k ; (c) parameter α ; (d) parameter β ; (e) parameter δ ; and (f) parameter fo Fig 12 Results of parameter identification: experimental data ( ), reconstructed... Current (A) (e) (f) Fig 11 Parameter identification results (−) and polynomial fitted coefficients (· · ·): (a) parameter c; (b) parameter k; (c) parameter α; (d) parameter β; (e) parameter δ and. .. conventional viscous damping and spring stiffness This approach, as an attractive feature, maintains a relationship between the damper parameters and physical force-velocity hysteretic phenomena and