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Bouc–Wen model-based real-time force tracking scheme for MR dampers

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IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 22 (2013) 045012 (12pp) doi:10.1088/0964-1726/22/4/045012

Bouc–Wen model-based real-time force

tracking scheme for MR dampersF Weber

Empa, Swiss Federal Laboratories for Materials Science and Technology, Structural EngineeringResearch Laboratory, Uberlandstrasse 129, CH-8600 Dubendorf, Switzerland

E-mail: felix.weber@empa.ch

Received 14 December 2012, in final form 13 February 2013

Published 11 March 2013

Online at stacks.iop.org/SMS/22/045012

Abstract

A Bouc–Wen model-based control scheme is presented which allows tracking the desired

control force in real-time with magnetorheological (MR) dampers without feedback from a

force sensor. The control scheme estimates the MR damper force by parallel computing of 

several Bouc–Wen models with different constant currents as inputs and for the actual MR

damper displacement and velocity, respectively. Based on the estimated forces and the desired

control force the MR damper current is determined by a piecewise linear interpolation scheme.

The model-based feed-forward control scheme is numerically and experimentally validated. If 

the desired control force is not constrained by the pre-yield region, residual force at 0 A and

force at maximum current, the very small force tracking error ≤0.0015 in the simulation is

caused by the control-oriented simplification of the linear interpolation scheme. The tests

reveal that the real-time control scheme is numerically stable and the force tracking error of 

≤0.078 represents an acceptable accuracy.

(Some figures may appear in colour only in the online journal)

1. Introduction

In recent years, controlled damping of civil structures with

magnetorheological (MR) dampers has been investigated bymany researchers and implemented on real civil structures.

A selection of such control systems can be found in [1–11].The main reasons for the use of MR dampers are that their

force range is suitable for civil structures; they are reliable,low-power consuming, seen to be fail-safe and can combine

the benefits of active and passive control systems [12, 13]. Thestructural damping that can be expected from semi-actively

controlled MR dampers depends on many aspects of thecontrol scheme, of which some of the relevant items are the

control law, number and location of MR dampers, numberand type of sensors, accuracy of observer and control force

tracking accuracy. The last item is determined by the controlforce tracking method that controls the MR damper current

in real-time for minimum difference between the actualMR damper force and the desired semi-active control force.

Existing control force tracking schemes for the real-time

control of the MR damper force may be divided into thefollowing classes.

(1) Model-based feed-forward control force tracking schemes.

(2) Nonlinear control force tracking schemes based on force

feedback.

(3) Combined feed-forward feedback control force tracking

schemes.

The approaches of the first class estimate the MR damper

current in real-time based on an MR damper model and thedesired control force and thereby avoid the use of a force

sensor. The Dahl model is adopted in [1, 6, 9, 14], the LuGrefriction model is adopted in [15, 16], neural network models

are developed in [7, 17, 18], fuzzy logic is used in [19]and a mapping approach in [8]. The Heaviside step function

approach with feedback from a force sensor belongs to the

second class [20]. The control force tracking schemes of thethird class combine a model-based feed-forward with a force

feedback to enhance the tracking accuracy [21–23].This paper presents a new approach to the first class of 

control force tracking schemes. The real-time force tracking

scheme is based on the Bouc–Wen model [20, 24–26] that is

used as observer and is extended by the Stribeck effect [27].The new approach is verified by simulation and the control

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Smart Mater. Struct. 22 (2013) 045012 F Weber

Figure 1. Normalized force response of the proposed Bouc–Wenmodel with Stribeck effect.

force tracking errors are discussed. It is shown that the errors

can be split into errors due to physical force constraints of MR

dampers and very small errors due to the proposed control

scheme. The Bouc–Wen model-based real-time force tracking

scheme is then experimentally validated. It is explained

how the Bouc–Wen model can be run numerically stable

in real-time in the Matlab R /dSPACE R environment. The

measured control force tracking accuracies are discussed and

conclusions are drawn.

2. Bouc–Wen model-based real-time force tracking

scheme

2.1. Bouc–Wen model including Stribeck effect 

The Bouc–Wen model proposed here is based on the original

approach and is extended by a second-order filter to model

the Stribeck effect [27]. If the MR damper does not include

an accumulator, as is the case for the rotational MR damper

described in [8, 9], the current-dependent MR damper force

 f (i) becomes

 f (i) = α(i) zStribeck  + c(i)˙ x (1)

where z

Stribeck 

denotes the evolutionary variable including theStribeck effect and ˙ x is the damper velocity. α(i) and the

viscous coefficient c(i) are modelled as functions of the MR

damper current i and not as functions of the command voltage

u to the current driver as done in, e.g. [23], because the current

driver of type KEPCO R that is used for the validation of 

the proposed real-time force tracking scheme is capable of 

almost fully compensating for the coil impedance of the MR

damper; hence, the actual current is almost equal to the desired

current at all instants (see section 4, figures 19 and 22). As a

consequence, current driver dynamics can be neglected. The

Stribeck effect is modelled by adopting the second-order filter

 Z Stribeck (s) = Z (s) ω20

s2 + 2ζ ω0s + ω20

(2)

where Z (s) is the complex evolutionary variable in the

frequency domain, s is the complex radial frequency, ω0 is the

radial frequency and ζ  the damping ratio of the second-order

filter. Figure 1 shows the force velocity trajectories of the

proposed Bouc–Wen model including the Stribeck effect.

The modelled MR damper force is normalized because the

proposed real-time force tracking scheme can be appliedto any MR damper with any maximum force capacity. It

is observed that the parameter settings of the second-order

filter generate the typical force overshoot due to the Stribeck 

effect [25–29]. The evolutionary variable z is governed by the

differential equation according to the Bouc–Wen approach

˙ z = −γ |˙ x| z| z|n−1 − β ˙ x| z|n + A˙ x (3)

where γ , β , n and A are parameters to shape the hysteretic

force part of the MR damper force. α(i) is modelled as a

quadratic function of the MR damper current as follows

α = αa + αb(i − 1.2imax)2 (4)

where αa and αb are shaping parameters and imax is the

maximum MR damper current. Approach (4) with negative

αb is able to model the nonlinear increase of the hysteretic

force part for increasing current, as is typically observed in

case of many MR dampers [29]. The nonlinear increase in the

hysteretic force part is visible in figure 1 from the fact that

the MR damper force increase from, e.g., 0.2imax to 0.4imax,

is greater than from, e.g., 0.8imax to imax. c(i) is modelled as a

linear function of current

c(i) = ca + cbi (5)

to take into account the effect that the MR damperviscous force part increases with increasing current.

This feature is visible in figure 1 from the increasing

slope of the force velocity trajectories in the post-yield

region [29]. The parameters of the proposed Bouc–Wen model

equation (1)–equation (5) are selected such that the simulated

force resulting from a sinusoidal displacement of 1 Hz and

50 mm shows the behaviour that is described above and is

typical for many cylindrical MR dampers. The parameters are

γ  = 200 1 m−1, β = 200 1 m−1, n = 2, A = 5000 1 m−1,

αa = 4250 N, αb = −27.7778 N A−2, ca = 1400 N s m−1,

cb = 1000 N s m−1 A−1, ω0 = 2π 25 rad s−1 and ζ  = 0.585.

The given units assume that z is a non-dimensional variable.

Please notice that the simulation of the proposed MR dampermodel does not target to model a specific MR damper but

is required in section 2.2 and later in section 3 to explain

the proposed Bouc–Wen model-based real-time force tracking

scheme.

2.2. Real-time force tracking scheme

Figure 2 shows the entire real-time force tracking scheme.

The desired control force f des due to the selected control

laws of viscous damping and clipped viscous damping with

negative stiffness is computed based on the actual MR damper

displacement x and velocity ˙ x. The desired control force,the displacement and velocity are input state variables to the

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Smart Mater. Struct. 22 (2013) 045012 F Weber

Figure 2. Bouc–Wen model-based real-time force tracking scheme.

Bouc–Wen model-based estimation of the current i. In order

to check the force tracking accuracy in simulation, the actual

MR damper force f act is then computed based on the estimated

current i, the actual displacement and velocity. For the case

under consideration where (1) does not depend on x due to

the absence of an accumulator as in, e.g. [23], (1) is computed

based on ˙ x and i only.

The core of the force tracking method is depicted infigure 2 by the close-up. The same Bouc–Wen model (1–5)

is computed in real-time for the constant currents i = 0 A,0.1imax, . . ., imax and for the actual x and ˙ x. The resultingforces f (0 A), f (0.1imax) , . . . , f (imax) represent the estimatedforces that would be generated by the MR damper if theapplied MR damper currents were i = 0 A , 0.1imax, . . .,imax. Based on the forces f (0 A), f (0.1imax) , . . . , f (imax) andthe desired control force f des, the interpolated current iint isderived by piecewise linear interpolation—here described forthe case when | f des| is larger than | f (0.2imax)| and smaller thanor equal to | f (0.3imax)|—as follows

iint =

0.2imax +(0.3imax − 0.2imax)(| f des| − | f (0.2imax)|)

 f (0.3imax) − f (0.2imax)

: | f (0.2imax

)| < | f des

| ≤ | f (0.3imax

)|

and sgn( f des) = sgn( f (0.3imax))

0 : otherwise.

(6)

The desired MR damper current ides takes the current

constraints into account, yielding

ides =

iint : 0 ≤ iint ≤ imax

0 : iint < 0

imax : iint > imax.

(7)

The force tracking accuracy resulting from the proposed

Bouc–Wen model-based real-time force tracking scheme is

validated by simulation and experiments, which are describedin sections 3 and 4.

3. Simulation results

The Bouc–Wen model-based real-time force tracking scheme

is adopted to track in real time the desired control forces due

to the control laws:

(a) linear viscous damping [9, 30], and(b) clipped viscous damping with negative stiffness [9, 31, 32]

with an MR damper that is modelled by the Bouc–Wen

approach with Stribeck effect (1–5). The control law (a)

is chosen as a test case because this benchmark damping

strategy is appropriate to explain the working behaviour of 

the proposed real-time force tracking scheme. The second

test scenario is the control law (b), which can double the

structural damping compared to linear viscous damping, since

it is a promising approach for semi-active structural control

and the limitations of the force tracking accuracy due to the

pre-yield stiffness and residual force at 0 A of MR dampers

can be demonstrated well. It is emphasized that the proposed

real-time force tracking scheme can be adopted to track any

desired control force in real time as given by, e.g., clipped

optimal control [2, 3], sliding mode control [33, 34], fuzzy

logic [35, 36] and neural network-based control schemes [17,

37, 38] that are often used for structural control.

3.1. Simulated real-time force tracking of viscous damping

The simulation of the system as depicted in figure 2

is made in Matlab R /Simulink R using the solver ode3

(Bogacki–Shampine) with a fixed time step size of 0.5 ms.The input to the Simulink R file is a sinusoidal displacement

with 5 mm amplitude and constant frequency of 1 Hz. The

desired control force as function of time t  is [9, 30]

 f des(t ) = cdes ˙ x(t ) (8)

where cdes is the desired viscous damping coefficient. The

simulated force tracking accuracies are shown in figure 3 by

the desired and actual force displacement trajectories and in

figure 4 by the desired and actual force velocity trajectories.

In addition to these trajectories, the trajectories resulting from

constant current are also plotted to better visualize the sources

of force tracking errors (FTEs). The simulated force tracking

is based on the following simplifications:

• undesirable current dynamics due to weak current drivers

do not exist, which is fulfilled as shown in section 4

(figures 19 and 22), and

• MR fluid dynamics due to the migration and aggregation

of particles do not exist [39].

As can be seen from figures 3 and 4, f des is not constrained by

 f (imax) due to the choice of  cdes = 4.85 × 104 N s m−1 and

the FTEs are mainly caused by:

• the residual force at 0 A of the MR damper, and• the pre-yield stiffness of the MR damper.

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Figure 3. Simulated tracking of viscous damping: forcedisplacement trajectories.

Figure 4. Simulated tracking of viscous damping: force velocitytrajectories.

Figure 5. Simulated tracking of viscous damping: close-up of forcedisplacement trajectories.

Figure 6. Simulated tracking of viscous damping: close-up of forcevelocity trajectories.

Figure 7. Simulated tracking of viscous damping: forces (a) anddesired current (b) versus time.

The close-ups in figures 5 and 6 and the figure 7 indicate how

the Bouc–Wen model-based real-time force tracking scheme

works.

(i) Within this section of the trajectory,f des can be fullytracked because f des is not constrained by f (0A), f (imax)

and the pre-yield stiffness. FTEs are not visible but exist

because the interpolation (6) is piecewise linear but the

hysteretic force part of the MR damper is a nonlinear

function of current (4). However, the small force intervals

used the interpolation (6), which corresponds to current

intervals of 0.1imax, end up in negligibly small FTEs,

i.e. | f des − f act|/| f des| ≤ 0.0015. Due to the nonlinear

relation between the hysteretic force of the MR damper

and current (4), the estimated current i does not describe

a pure half-sine as a function of time (figure 7(b)).

(ii) Within this section of the trajectory, f des < f (0 A) and

sgn( f des) = sgn( f (0A)). The minimal FTE is achieved if i = 0 A is applied, which is realized by the real-time force

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Figure 8. Simulated tracking of viscous damping constrained bymaximum MR damper force: force displacement trajectories.

tracking scheme, see figure 7. Consequently, f act tracks

 f (0A). The FTE here is given by the residual force at 0 A

of the MR damper.

(iii) Within this section of the trajectory, sgn( f des)     =

sgn( f (0A)) due to the pre-yield region. Due to

sgn( f des)     = sgn( f (0A)), i = 0 A minimizes the FTE,

which is realized by the real-time force tracking scheme.

(iv) This section starts where the pre-yield region ends and

thereby sgn( f des) = sgn( f (0A)). From figures 5 and 6 it

can be observed that the shortest trajectory to get back 

to f des is to follow the trajectory resulting from imax.

The Bouc–Wen model-based real-time force trackingapproach exactly does this: the estimated current steps

from i = 0 A to imax (figure 7(b)) in order to come back to

 f des as fast as possible (figure 7(a)) and thereby minimize

the FTE.

(v) As in section (i), the tracking of  f des is not constrained

by f (0A), f (imax) and the pre-yield stiffness and the

resulting very small FTE of  | f des − f act|/| f des| ≤ 0.0015

is caused only by the linear piecewise interpolation (6).

3.2. Simulated tracking of viscous damping constrained bymaximum MR damper force

The Bouc–Wen model-based real-time force tracking scheme

is also checked for the case when f des is constrained by f (imax)

due to a larger cdes, i.e. 1.5 × 4.85 × 104 N s m−1. Figures 8,

9 and 10(a) plot the resulting force tracking accuracies. In

addition to the FTEs as described in section 3.1, FTEs due

to the maximum MR damper force limitation f (imax) occur.

As can be seen from figure 10(b), the proposed real-time

force tracking scheme outputs i = imax if | f des| > | f (imax)| and

sgn( f des) = sgn( f (imax)) due to (6) and (7).

The simulations presented in sections 3.1 and 3.2 show

that the sources of FTEs can be split into physical force

constraints of MR dampers and inaccuracies of the real-time

force tracking method under consideration. Neglecting thedynamics of the current driver and MR fluid, the physical

Figure 9. Simulated tracking of viscous damping constrained bymaximum MR damper force: force velocity trajectories.

Figure 10. Simulated tracking of viscous damping constrained bymaximum MR damper force: forces (a) and desired current (b)versus time.

force constraints of MR dampers besides their semi-active

restriction are:

• residual force at 0 A,• maximum force at imax, and

• pre-yield stiffness.

The FTEs due to the proposed Bouc–Wen model-based

real-time force tracking scheme:

• are caused by the piecewise linear interpolation in (6),

which is a reasonable, control-oriented simplification of 

the fact that the relation between the hysteretic MR damper

force and current is nonlinear (4), and

• are very small, i.e. | f des − f act|/| f des| ≤ 0.0015, due to the

small force intervals, which correspond to current intervalsof 0.1imax, that are used in the interpolation scheme (6).

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Figure 11. Simulated tracking of clipped viscous damping withnegative stiffness: force displacement trajectories.

Figure 12. Simulated tracking of clipped viscous damping withnegative stiffness: force velocity trajectories.

3.3. Simulated tracking of clipped viscous damping withnegative stiffness

Viscous damping with negative stiffness is a preferable

control law in structural control because the negative stiffness

force increases the structural flexibility at the control deviceposition [9, 31, 32]. This enlarges the motion and thereby the

energy dissipation in the control device, which finally leads

to augmented structural damping. If this control law is to be

emulated with MR dampers, the common method is to clip the

active desired control forces to zero. This yields the desired,

semi-active control force as follows

 f des(t ) =

cdes ˙ x(t ) − k des x(t )

: ˙ x(t )(cdes ˙ x(t ) − k des x(t )) ≥ 0

0 : ˙ x(t )(cdes ˙ x(t ) − k des x(t )) < 0

(9)

where k des is the desired stiffness coefficient. The chosen

parameters for the numerical test are cdes = 2.909 ×104 N s m−1 and k des = −2 × 105 N m−1, where cdes is

Figure 13. Simulated tracking of clipped viscous damping withnegative stiffness: forces (a) and desired current (b) versus time.

reduced compared to viscous damping, only that f des is notconstrained by | f (imax)|. As can be observed from figure 11,

 f des describes an ellipse with negative inclination due to thenegative stiffness term. The active desired control forces of 

viscous damping with negative stiffness that are plotted by

the dashed blue line are clipped to zero, which evokes thehorizontal and vertical trajectory parts of  f des. As indicated

by the arrows in figure 12, the negative stiffness force leadsto a force velocity trajectory that turns clockwise in the

post-yield region. Looking at figures 11–13, the following canbe observed.

(i) Within this section of the trajectory, f des is not constrainedby f (0A), f (imax) and the pre-yield stiffness. The

resulting FTE due to the proposed real-time force

tracking scheme is caused by the linear piecewiseinterpolation in (6) and is | f des − f act|/| f des| ≤ 0.0015.

(ii) Within this section of the trajectory, f des < f (0A) and

sgn( f des) = sgn( f (0A)). The FTE is minimized by i =

0 A, which is applied by the real-time force tracking

scheme.

(iii) In this section the MR damper is operated in the pre-yieldregion and sgn( f des) = sgn( f (0A)). Hence, i = 0 A is the

best choice to minimize the FTE.

(iv) Within this section of the trajectory, the MR damperis still operated in the pre-yield region and sgn( f des)     =

sgn( f (0A)) f des due to the jump of  f des. As for section(iii), i = 0 A minimizes the FTE which is realized by the

real-time force tracking scheme.

(v) This section starts where the pre-yield region ends and

thereby sgn( f des) = sgn( f (0A)). The shortest trajectoryto get back to f des is given by the trajectory resulting from

imax. As visible in figure 13, the real-time force trackingscheme outputs the current step from i = 0 A to imax and

reduces i very rapidly when f des is reached in order totrack f des as in section (i).

(vi) As in section (i), the FTE is only given by the real-timeforce tracking method under consideration.

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Smart Mater. Struct. 22 (2013) 045012 F Weber

Figure 14. Cable damper setup with measurement and control hardware.

The force tracking analysis of clipped viscous damping with

negative stiffness shows that:

• the pre-yield stiffness at imax limits how accurate the jump

in f des can be tracked by f act, and

• the active desired control forces are constrained, first, by

clipping and, second, by the residual force at 0 A.

4. Experimental results

4.1. Experimental setup

The proposed real-time force tracking scheme is experi-

mentally validated on a steel wire strand with a rotational

MR damper positioned at 4% of the cable length from

the left anchor (figure 14). Detailed information on the

rotational MR damper under consideration can be found

in [40]. The cable properties are: the tensile force is 22 kN,

the mass per unit length including the additional clamped

masses is 5.85 kg m−1 and the length is 16.54 m. The

displacement at the MR damper position is measured by

a laser triangulation sensor and the cable acceleration is

recorded by an accelerometer. The actual MR damper force

is measured by a 500 kN load cell to determine the FTE.

The desired MR damper current is produced by the current

driver of type KEPCO R which almost fully compensates for

the coil impedance. As a result, the time lag and difference

between the desired and actual currents are negligibly small,

see figures 19 and 22.

The Bouc–Wen model-based real-time force tracking

scheme is implemented in the Matlab R /dSPACE R environ-

ment and runs at 2000 Hz sampling frequency in real time.

The collocated velocity is estimated from the measured

displacement and acceleration by adopting a kinematic

Kalman filter [9]. Numerical instabilities of the Bouc–Wen

model-based force tracking scheme due to measurement

noise in the input states are compensated by the follow-ing limitations: min(˙ z) = −2000, max(˙ z) = 2000, min( z) =

−1, max( z) = 1. These limitations are required only when

the cable is almost non-vibrating. If the cable amplitudes are

larger than 0.2 mm at the MR damper position, the limitations

of  ˙ z and z do not constrain ˙ z and z, respectively. Notice that

 z is a non-dimensional variable with an operating range of 

[−1, 1]. To make z less susceptible to measurement noise in

the input states, the evolutionary variable z is filtered by a low

pass of first order (−20 dB/decade) with corner frequency

300 Hz.

4.2. Validated Bouc–Wen model of rotational MR damper 

First, the force responses of the rotational MR damper

resulting from constant currents was measured using a

hydraulic actuator of type Instron R that imposed a sinusoidal

displacement of 8 mm amplitude at 1.5 Hz (figures 15 and 16).

The measured trajectories show the following features.

• The pre-yield stiffness increases with increasing currents,

as typically observed for many MR dampers and also seen

in the simulated force displacement trajectories due to

constant current in section 3 (figures 5 and 6).

• The viscous force part of the MR damper under

consideration that can be read off from the force velocity

trajectories between the Stribeck effect and ˙ x = 0 is very

small [29].

• The force response between the pre-yield and post-yield

regions shows an almost linear and fairly slow increase

as a function of the MR damper displacement and time,

respectively. This slow force response results from the

fact that, when the disc starts to rotate sinusoidally in one

direction (starting from max(− x)), first, the particle chains

at the outer diameter of the disc enter the post-yield regime

due to the larger shear deformation at this position [40,

41]. Then, with increasing rotation angle and MR damper

displacement, respectively, also particle chains that are

located closer to the centre of the disc enter the post-yieldregime. Finally, when the rotation of the disc corresponds

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Smart Mater. Struct. 22 (2013) 045012 F Weber

Figure 15. Validated Bouc–Wen model with Stribeck effect: forcedisplacement trajectories.

Figure 16. Validated Bouc–Wen model with Stribeck effect: forcevelocity trajectories.

to approximately 2 mm of MR damper displacement

for the case under consideration, all particle chains are

operating in the post-yield regime and consequently the full

post-yield force is produced.

The parameters γ  = 10000 1 m−1

, β = 10000 1 m−1

,n = 2 and A = 15000 1 m−1 are selected by trial and error

to obtain the same slope of the simulated force displacement

trajectories in the pre-yield region as measured. The nonlinear

relation between the hysteretic force part and current (4) is

modelled by a piecewise linear curve fit and implemented

in Matlab R /dSPACE R by a one-dimensional look-up table.

The viscous force part is estimated from the measured force

velocity trajectories in the post-yield region. The resulting

parameters are ca = 34 N s m−1 and cb = −8 N s m−1 A−1.

The parameters of the second order filter to model the Stribeck 

effect are chosen to be ω0 = 2π 12 rad s−1 and ζ  = 0.78 to

obtain the same force overshoot during the same displacement

range as seen in the measurements. Since the approach (1–5)does not take into account the effect that the post-yield

Figure 17. Measured tracking of viscous damping: forcedisplacement trajectories.

Figure 18. Measured tracking of viscous damping: force velocitytrajectories.

regime is a function of the rotation angle as described above,

the simulated and measured trajectories show a discrepancy

where the MR damper force leaves the pre-yield regime and

enters the post-yield regime (figures 15 and 16). In contrast,the model shows a very good accuracy in the pre-yield region

and in the part of the post-yield region where all particle

chains are operated in the sliding regime.

4.3. Measured tracking of viscous damping

The proposed real-time force tracking scheme is first tested by

tracking f des resulting from viscous damping (8) with cdes =

2000 N s m−1. This choice of cdes guarantees that the desired

force is not constrained by | f (imax)|. The measured force

tracking is plotted in figures 17 and 18 during two periods of 

vibration of a free decay response of the cable. It is observedthat max( f des) is smaller than max(− f des). The reason for this

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Figure 19. Measured tracking of viscous damping: desired andactual currents versus time.

is that the cable peak velocity at the MR damper position

is greater when the cable swings downwards than upwards.

FTEs are mainly visible in sections (i) and (ii).

(i) Section (i) denotes the location in the trajectory where the

MR damper force gets out of the pre-yield regime and

enters the post-yield regime, where the desired force is

not constrained by | f (0A)| and | f (imax)|. Due to | f des| <

| f (0A)| in the pre-yield region, the MR damper locks

the cable. When the desired force can be tracked in the

post-yield region, the MR damper starts to rotate. The

transition between the operating condition ‘locked’ and‘rotating’ induces higher frequency components in the

collocated cable displacement, which are then also present

in the collocated velocity ˙ x and consequently in f des, due

to the control law f des = cdes ˙ x.

(ii) The oscillations visible in section (ii) represent the

transient closed-loop response of the initially triggered

higher frequency components in section (i). The mean

FTE within the trajectory part from point ‘a’ to point

‘b’, where the force tracking is not constrained by the

pre-yield stiffness, f (0A) and f (imax), is 0.0780. This

value is of the same order as the measured FTEs presented

in [23, 42] and therefore can be seen as acceptably small,considering that the MR damper force is controlled in

real-time by a model-based feed-forward without force

feedback from a sensor.

The desired and actual currents are depicted in figure 19,

which demonstrate that the current driver of type KEPCO R is

capable to almost fully compensate for the coil impedance of 

the MR damper under consideration. Hence, the modelling

approach in section 2 that does not take into account current

dynamics due to insufficiently powerful current drivers is

 justified. The current time history shows the typical peaks

to imax when the Bouc–Wen model-based real-time force

tracking scheme targets to reach f des as fast as possible afterthe pre-yield region, see (iv) in section 3.1. After this current

Figure 20. Measured tracking of clipped viscous damping withnegative stiffness: force displacement trajectories.

peak, the commanded current generates approximately the

correct actual force and becomes zero when | f des| < | f (0A)|.

The higher frequency components in the desired current,

which mainly take place immediately after the current peak,

are caused by the higher frequency components in f des due

to the transition from the pre-yield region to the post-yield

region. The different current maxima when f des is tracked in

the post-yield region correspond to the fact that max( f des) is

smaller than max(− f des) due to the larger peak velocity of the

cable when it swings downwards than upwards.

4.4. Measured tracking of clipped viscous damping withnegative stiffness

In order to guarantee that the desired force is not constrained

by the maximum MR damper force at 3.5 A, i.e. | f des| <

| f (imax)|, cdes is reduced compared to viscous damping only

because the stiffness term in (9) increases the maximum

desired force significantly. The desired parameters in (9)

for the test under consideration are chosen as cdes =

1150 N s m−1 and k des = −28000 N m−1. The resulting

force displacement and force velocity trajectories during two

periods of vibration during the free decay response of the

cable are plotted in figures 20 and 21 and the correspondingdesired and actual currents are displayed in figure 22. Itis seen

that:

(i) The current spike within the section (i) forces f act to

follow the trajectory due to imax and thereby minimizes

the FTE. The resulting FTE is given by the finite

pre-yield stiffness of the force displacement trajectory

at imax. Only the hypothetical case of an infinite large

pre-yield stiffness due to infinite large imax would allow

tracking the jump in f des by f act without FTEs.

(ii) During the section (ii), f act < f des despite the current is

still i = imax. Here, f des cannot be precisely tracked due

to the effect that the full development of the post-yieldforce requires a rotation in the disc that corresponds

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Smart Mater. Struct. 22 (2013) 045012 F Weber

Figure 21. Measured tracking of clipped viscous damping withnegative stiffness: force velocity trajectories.

to approximately 2 mm damper displacement. This

characteristic of the MR damper under consideration is

described in section 4.1, where the full force response in

the post-yield region also requires approximately 2 mm

damper displacement.

(iii) In this section, the force tracking is not constrained by

 f (0A), f (imax), the post-yield region and the dependency

of the post-yield force on the rotation angle as in section

(ii). The mean FTE due to the real-time force tracking

scheme within the trajectory part from point ‘a’ to point

‘b’ is 0.0569. Compared to the measured FTEs obtained

in [23, 42], this value represents a fairly small value forthe real-time controlled MR damper force without force

feedback from a sensor.

(iv) In this section, first f act > f des due to f (0A) and then f act >

 f des because of the clipping. The real-time force tracking

scheme commands the correct current for minimum FTE,

which is i = 0 A. The reason why f act does not track 

 f (0A) from the very beginning of this section although

i = 0 A is applied are remanent magnetization effects

in the particles due to the magnetization in the previous

sections (i)–(iii) [23]. The remanent magnetization

evokes a slightly larger force at i = 0 A than f (0A) that

results from the application of  i = 0 A for a longer timeas, e.g., when measuring the MR damper characteristics

at constant current [23], see section 4.2, figure 15.

5. Summary and conclusions

A new approach to track the desired control force in real-time

with MR dampers is presented. The approach is based on a

Bouc–Wen model-based feed-forward without feedback from

a force sensor. The Bouc–Wen model is adopted to estimate

the MR damper force in real-time for the actual MR damper

displacement and velocity, respectively, and as if the constant

currents of  i = 0 A , 0.1imax, . . ., imax were applied. Theresulting estimated force that would be generated by the MR

Figure 22. Measured tracking of clipped viscous damping withnegative stiffness: desired and actual currents versus time.

damper if the constant currents of  i = 0 A, 0.1imax, . . ., imax

were applied together with the desired control force are then

used to estimate the desired current in real-time by a linearpiecewise interpolation scheme.

The model-based feed-forward control scheme is tested

by simulation to track linear viscous damping and clippedviscous damping with negative stiffness. These two control

laws are selected because the working behaviour of the

proposed force tracking scheme can be shown well. However,it is emphasized that the proposed force tracking scheme

can be adopted to track any desired control force resulting

from, e.g., clipped optimal control, sliding mode control,fuzzy logic, neural network-based schemes and others. It

is demonstrated that the proposed real-time force trackingscheme is able to track the desired control force in both cases

with a tracking error ≤0.0015 if the desired control forceis not constrained by the physical force constraints of MR

dampers, which are pre-yield stiffness, residual force at 0 A

and maximum force at imax. It is explained why a current peak minimizes the force tracking error when the MR damper is

operated in the pre-yield region and the signs of the desiredcontrol force and the pre-yield MR damper force are equal.

The force tracking scheme is experimentally validated on

a rotational MR damper that is connected to a vibrating stealwire strand. It is explained how the Bouc–Wen model is run

in real time in the Matlab R /dSPACE R environment without

numerical instabilities. The experimental results show that themodel-based feed-forward force tracking scheme is able to

track the two control laws under consideration without anynumerical instability and with an average force tracking error

of 0.057–0.078. It is therefore concluded that the proposed

control scheme is a reliable and efficient force tracking toolfor the real-time control of MR dampers.

Acknowledgments

The author gratefully acknowledges the financial supportof Empa, Swiss Federal Laboratories for Materials Science

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Smart Mater. Struct. 22 (2013) 045012 F Weber

and Technology, Dubendorf, Switzerland, and the technical

support of the industrial partner Maurer Sohne GmbH and Co.

KG, Munich, Germany.

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