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 ParticleSwarm Optimization N.. A new model for MR dampers is proposed in this paper.For t

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 ical (MR) fluid dampers A new model for MR dampers is proposed in this paper.For this, computationally-tractable algebraic expressions are suggested here in con-trast to the commonly-used Bouc-Wen model, which involves internal dynamicsrepresented by a nonlinear differential equation In addition, the model parame-ters can be explicitly related to the hysteretic phenomenon To identify the modelparameters, a particle swarm optimization (PSO) algorithm is employed using ex-perimental force-velocity data obtained from various operating conditions In our

magnetorheolog-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 Parameteridentification results are included to demonstrate the accuracy of the model and theeffectiveness 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)

Although the MR damper is promising in control applications, its major back 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

draw-be challenging in the case of employing the MR damper The modelling of thehysteresis had been studied in [5] and [6] including the Bingham visco-plasticmodel, the Bouc-Wen model, the modified Bouc-Wen model and many others.These models range from simple dry-friction to complicated differential equa-tion representations However, it is noted of a trade-off between the modelaccuracy and its complexity From the control engineering point of view, non-linear differential equation based models may affect robustness of the controlsystem and hamper the feasible controller realization as in the case of MRdampers employed in the reduction of seismic response in buildings [7]

On the other end of the spectrum, polynomial based modelling of the MRdamper was attempted in [8] with a reduction in model complexity A hyper-bolic function based curve-fitting model was proposed in [9] with satisfactoryresults A black-box damper model was also applied in [10] as an alternative.Soft-computing techniques, fuzzy inference systems and neural networks werealso applied in modelling a MR damper; see [11] and [12], which representanother paradigm for a suitable approach towards an efficient model Evo-lutionary computation methods, e.g., genetic algorithms [13], [14], have alsobeen widely applied in modelling and parameter identification applicationsand many others In [15], the genetic algorithm (GA) was employed to iden-tify a mechatronic system of unknown structure However, in addition to theimplementation complexity, the GA may found difficulties in convergence to

optimal parameters unless elitism [16] is explicitly incorporated in the rithm An identification procedure following this approach was reported in[17] Moreover, a priori knowledge on the ranges of solutions may be required

algo-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 modelparameter identification when the identification is cast as an optimizationproblem The PSO is based on the multi-agent or population based philosophy(the particles) which mimics the social interaction in bird flocks or schools offish, [18] By incorporating the search experiences of individual agents, thePSO is effective in exploring the solution space in a relatively smaller number

of iterations, see [19] In emulating bird flocks, particles are assigned withvelocities that lead their flight across the solution space The best solutionsfound by an individual particle and by the whole population are memorizedand used in guiding further search flights The best solution is reported at thesatisfaction of some termination criteria Other applications of the PSO can

be found in the design of PID controllers [20] and electro-magnetics [21] ThePSO convergence characteristic was analyzed in [22], where algorithm controlsettings were also proposed

In this work, a novel model for the MR damper will be proposed This modeluses a hyperbolic tangent function to represent the hysteretic loop togetherwith components obtained from conventional viscous damping and spring stiff-ness This approach, as an attractive feature, maintains a relationship betweenthe damper parameters and physical force-velocity hysteretic phenomena andreduces the complexity from using a differential equation to describe the hys-teresis The PSO is then applied to identify the model parameters using exper-imental force-velocity data obtained from a commercial MR damper Here, toenhance 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 identifiedparameters

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 issuggested in Section 3 with a discussion of the model parameters The PSO, asapplied in identifying the proposed model parameters, is presented in Section

4 and its advantages are highlighted Control settings for the algorithms aredetermined and a performance-enhanced criterion is also proposed Results ofparameter identification are given in Section 5 together with some discussion.Finally, a conclusion is drawn in Section 6

2 Magnetorheological Damper

2.1 Principle of Operation

The magnetorheological damper may be viewed as a conventional hydraulicdamper except that the contained fluid is allowed to change its yield stressupon the application of an external magnetic field The structure of the damper

is sketched in Fig 1

Fig 1 MR damper structure

Tiny magnetizable particles, e.g., carbonyl iron, are carried in non-magneticfluids 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 MRfluid changes inversely to the temperature and a reduction in the damper forceresults with an increase in temperature However, an appropriately designedcurrent controller can be applied to compensate for the change in the damperforce This controller will also be employed for counteracting the long-termstability problem of the MR fluid The MR damper used in this work is a RD-1005-3 model manufactured by the LORD Corporation The damper has acompressed length of 155mm, weighs 800g, accepts a maximum input current

of 2A at 12V DC and the response time is less than 25msec Interested readersare referred to the product information provided by the manufacturer [24]

Fig 2 MR damper test gear.

A sinusoidal excitation of small magnitude (e.g., 4mm-12mm) and at lowfrequencies (e.g., 1Hz-2Hz) is applied from a hydraulic drive to the damper asthe damper displacement The damper forces generated, under the application

of a set of magnetizing currents (0-2A) are measured by a force sensor (loadcell) mounted on the upper end of the damper The displacement and thedamper force readings are recorded for parameter identification A typicaldamper force plot is depicted in Fig 3 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 cous damping and spring stiffness coefficients respectively This results in anelliptical plot for the force/displacement relationship and an inclination of theellipse in the force/velocity response However, due to non-linearity, there arediscontinuities occurring at the extremes of the damper piston stroke travellimits of ±8mm, in our experiments The discontinuity also appears in a lag-ging effect of the force with respect to the velocity, within the ±20mm/srange, and produces the hysteretic phenomenon as shown Our experimentswere conducted at room temperature, around 25◦

vis-C The study on the perature effect, which has certain influences on the MR fluid properties [23],

tem-is out of the scope of thtem-is research

Velocity(mm/s)

0.25A 0.50A 0.75A 2.00A

(b)Fig 3 Characteristics of damper force vs supply current: (a) non-linearity inforce-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 commonlyemployed in previous work

2.3.1 Bingham Model

The stress-strain visco-plastic behaviour is used in the Bingham model Themodel contains a dead-zone or a discontinuous jump in the damper force/velocityresponse The damper force is expressed as

where f is the damper force, fc is the magnitude of hysteresis, sign(·) is thesignum function, ˙x is the velocity, x is the displacement, c0 is the viscouscoefficient 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 whenthere 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 teretic component The model can be described by the force equation and theassociated hysteretic variable, given by

2.3.3 Modified Bouc-Wen Model

In the modified Bouc-Wen model, additional parameters are introduced inorder to obtain a more accurate model It is given as

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 themodelling accuracy However, the model complexity is unavoidably increasedwith 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

A simple model is proposed here to model the hysteretic force-velocity acteristic of the MR damper A component-wise additive strategy is employedwhich contains the viscous damping (dashpot), spring stiffness and a hystereticcomponent, Fig 4 illustrates the conceptual configuration

char-Fig 4 Hysteresis model - component-wise additive approach.

3.1 Mathematical Model

In terms of mathematical expressions, the model makes use of a hyperbolictangent function to represent the hysteresis and linear functions to representthe viscous and stiffness The model is given by

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 tangentfunction 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 sequent inclusion in controller design and implementation A description and

sub-an sub-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 5 which lustrates the effects of the parameters on the damper force-velocity response

il-The viscosity c ˙x generates an inclined line that represents the post-yield (at thetwo ends of the hysteresis) relationship between the velocity and the damperforce Large coefficient c gives a steep inclination The stiffness k, (the hor-izontal ellipse formed from the product kx) is responsible for the openingfound from the vicinity of zero velocity A large value of k corresponds to theopening of the ends Parameters c and k contribute to the representation of aconventional damper without hysteresis

c k

β

0

Final Hysteresis

Fig 5 Hysteresis parameters

The basic hysteretic loop, which is the smaller one shown in Fig 5, is mined by β This coefficient is the scale factor of the damper velocity definingthe hysteretic slope Thus a steep slope results from a large value of β Thescale factor δ and the sign of the displacement determine the width of thehysteresis through the term δsign(x), a wide hysteresis corresponds to a largevalue of δ The overall hysteresis (the larger hysteretic loop shown in Fig 5)

deter-is scaled by the factor α determining the height of the hysteresdeter-is The overallhysteretic loop is finally shifted by the offset f0

After the development of the simple model, we proceed to identify the modelparameters and the approach adopted will be detailed in the following sections

4 Enhanced Particle Swarm Optimization

The particle swarm optimization (PSO) is inspired by the social interactionand individual experience [18], observed through human society developmentand natural behaviours of bird flocks and fish schools This technique is a pop-ulation based and controlled heuristic search and has been applied in a widedomain in function optimizations, [19] In the following, the PSO is comparedwith its counterpart, the genetic algorithm (GA), to highlight its potential inreduction 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 rithm inspired by natural evolution It is widely applicable, for example, in

algo-system identification, control, planning and scheduling and many others, see[13] and the references therein A basic GA can be described by the pseudocode 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 Schematatheory [14], the quality of the solution is a trade-off between accuracy and com-putational 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., ploitation, through the selection and crossover operator The search ability isenhanced through the use of the mutation operator with regard in exploration.However, the exploration process is generally slow and this may increase thesearch time span when the initial population does not cover the solution How-ever, in practice, a priori knowledge on the solution that can be used to guidethe initialization of the population may not always be available Furthermore,the crossover operator may destroy useful solutions Hence, an elitism strategy

ex-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 birdflights That is, each particle is assigned a velocity to update its value as a newposition 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) (9)

where vkis the present particle velocity, ω is the velocity scale factor, c1 and c2are uniform random numbers c1 ∈ [0, c1 ,max] and c2 ∈ [0, c2 ,max], xg is the groupbest (global) solution found up to the present iteration, xp is the personal bestsolution found by individual particles from their search history through timeindex 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

A comparison on the computational efficiency between GA and PSO mayreveal that the later is more attractive Within a single iteration in the two al-gorithms, the evaluation of the fitness of each potential solution is mandatory

in both algorithms The selection and crossover operator in GA are two-passoperations while a small number of mutation operations are conducted Onthe other hand, the finding of group and personal best are single-pass opera-tions The updates of velocity and position are simple additions Furthermore,the generation of random numbers, crossover/mutation operation in GA andscaling by c1 and c2 in PSO are common to both algorithms with similarcomplexities The saving in PSO computation is obtained from its simplifiedcalculations for the group- and personal-best particle In addition, PSO inher-ently 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 analysisusing control system theories, see [22] The confidence on the optimizationresult 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 theposition error of a particle as

Following Eq 9 and 10, the particle position error and velocity can be written

in the state-space form as

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 → 0 and

vt → 0 as time t → ∞ When the best solution is found xg becomes a constantand xp will tend to xg if the system is stable The stability of a discrete controlsystem can be ascertained by constraining the magnitude of the eigenvalues

λ1 ,2 of the system matrix A ∈ R to be less than unity, that is

λ1 ,2∈ {λ|λ2

− (1 + ω − c1− c2)λ + ω = 0}, |λ1 ,2| < 1 (14)

By choosing the maximal random variables c1 and c2 to be c1 ,max = 2 and

c2 ,max = 2 and take the expectation values from a uniform distribution, thecoefficients become c1 = 1 and c2 = 1 This case corresponds to a total feed-back of the discrepancy of the particle positions from the desired solution at

xg Writing out the eigenvalues, we have

4.3 Enhanced Termination Criteria

The PSO is basically a recursive algorithm that iterates until some terminationcriteria is met In common practice, the termination criteria may be defined

as the expiry of a certain number of iterations An alternative strategy usuallyemployed 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 tween a null hypothesis and an alternative The action according to the nullhypothesis is to continue the iteration, while the alternative action is to ter-minate but with a specific bound on the decision error The hypothesis can

be-be stated as

H0: there will be improvements in the fitness (16)

H1: there will be no improvement in the fitness (17)

From the structure of the PSO, it is noticed that the algorithm explicitlymaintains the group-best xg with the associated minimum fitness (e.g., where

a minimization problem is considered) For a minimization problem as sidered here, the fitness, f it, corresponding to xg will not increase as the