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Modelling, parameter identification, and controlof a shear mode magnetorheological deviceR Russo and M Terzo*
Department of Mechanics and Energetics, University of Naples Federico II, Naples, Italy
The manuscript was received on 12 November 2010 and was accepted after revision for publication on 25 January 2011.
DOI: 10.1177/0959651811400521
Abstract: This paper describes theoretical and experimental studies of a shear mode magne-
torheological fluid device. A testing procedure is used to measure the fundamental quantitiesneeded to define a state space non-linear model and to identify its parameters. The obtainedmodel is then validated by means of comparisons between modelling results and measure-ments. Finally, the model is used in the development of a time domain control scheme to reg-ulate the transmitted torque.
Keywords: magnetorheological fluid, state space model, non-linear optimal control
1 INTRODUCTION
Magnetorheological (MR) fluids consist of suspen-sions of micron-sized ferrous particles immersed in a
carrier fluid. They are characterized by the property
of being able to change their rheological characteris-
tics as a function of an applied magnetic field. This
property makes them of interest for use in control
devices for mechanical systems.
The literature on MR fluids dates back to the late-
1940s [1] with a considerable increase in research
activity being apparent in the last 20 years as they
became commercially available. Several MR fluid-
based smart devices have been described in the recent
literature including vibration dampers, clutches, andbrakes [27]. MR fluids are generally used in one of
two different modes: flow mode (fluid flowing through
an orifice) or shear mode (fluid shearing between two
surfaces that are in relative motion).
This paper presents a theoretical and experimen-
tal study on a shear mode MR device. The designed
and developed device was experimentally examined
on a test rig in order to acquire the data needed for
system modelling, parameter identification, model
validation, and control design. Devices similar to
the one discussed in this paper have been modelled
as first-order linear systems in the literature [8, 9].
However, the collected experiment data suggestedthat a state space first-order non-linear model
would be a better choice to simulate this type of sys-
tem. This option was investigated in the presented
simulation and control system development studies.
The so-defined model was adopted in the parameter
identification procedure and in non-linear optimal
control development based on the state-dependent
Riccati equation (SDRE).
2 DEVICE AND TEST RIG DESCRIPTION
2.1 The device
The device prototype used in the experiments was
developed based on the principles of multi-plate
viscous coupling. With reference to Fig. 1, the device
consists of two series of discs (A); one series being
integral with an internal rotor (B) and the other
integral with a second internal rotor (C). The two
rotors are held in relative motion by means of suit-
able bearings (B1). The discs are separated by
spacer elements (D) in order to create a gap, which
can be seen in the expanded view insert in Fig. 1,which is filled with the MR fluid. The shear mode
flow takes place in the gap between the surfaces
*Corresponding author: Department of Mechanics andEnergetics University of Naples Federico II , Italy.
email: m.terzo@unina.it
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that are in relative motion. The magnetic field direc-tion is at right angles to the direction of flow.
An external fixed casing (E) contains slots (F) for
30-coil series bobbin placement.
The considered device could be used as a brake or
clutch. It can also act as an internal friction torque
source if integrated in a semi-active automotive dif-
ferential [10]. The characteristics of the device are
listed in Table 1.
A careful selection of materials was made in order
to obtain MR fluid magnetization. Silicon steel was
used for the magnetic circuit components whereas
aluminium was used for the non-magnetic parts.The adopted MR fluid was MRF 132DG (Lord
Corporation, North Carolina, USA).
2.2 Test rig
A test rig (Fig. 2) was built to perform the experi-
mental investigations. It includes the MR device (A),
an inverter-piloted electrical motor (B), and a dyna-
mometric brake (C). In the experiments the brake
was characterized by a rotor that was integral with
the stator in order to create a pure fixed end.
Therefore, the device was tested as brake, with itsoutput rotor (B in Fig. 1) being held fixed.
The measured quantities were:
(a) the rotational speed of the electrical motor
which was measured by a phonic wheel and
proximity pick up;
(b) the input current which was measured by a
Hall effect closed-loop current sensor;
(c) the temperature of the device which was mea-
sured by an infrared thermometer sensor;
(d) the transmitted torque which was measured bya high stiffness strain gauge load cell.
The position of the infrared thermometer sensor is
shown in Fig. 1. The temperature was measured on
the flange surface of the MR device and was
employed as an indication of the MR fluid tempera-
ture. The system was powered by an adjustable 3
kW d.c. power supply. All measured quantities were
acquired and stored by a digital oscilloscope.
3 MATHEMATICAL MODEL
The device is classified as a multi-input single-out-
put device. The inputs are constituted by the supply
current and rotational speed and the output is the
transmitted torque. The viscoplastic behaviour of
the MR fluid can be modelled using Binghams law
tsdgdt , H
= ty H +h
dgdt ts. ty
dgdt = 0 ts\ty
((1)
where ts is the shear stress, h is the fluid viscosity,dg=dt is the shear rate, H is the magnetic field, and
ty the yield stress.
According to the Bingham model, the transmitted
torque is produced by two different contributions.
The first one (Tv) consists of a viscous torque
(Newtonian contribution) and the second one (Tm)
depends on the magnetization of the MR fluid.
The whole device can be considered in terms of
consisting of two branches, constituted by the rota-
tional speed and current, respectively (Fig. 3).
The rotational speed is the difference between
the input and output speeds of the device. Since thedevice is being considered as a brake the output
speed is taken to be zero.
Fig. 1 Device section and magnetic flow path
Fig. 2 The test rig
Table 1 Characteristics of the device
Number of gaps containing MR fluid 40
Gap outer radius 52 mmGap inner radius 33 mmGap height 1 mm
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Several experimental tests, performed in the range
0100 rad/s with no current input, showed that the
relation between torque and velocity was purely
algebraic, i.e. torque followed speed without any
appreciable lag. Consequently, in accordance with
the literature [11, 12], the first branch, related to
rotational speed input, was modelled using the
Newtonian viscous law.
At the same time, the tests highlighted a depen-
dence of the viscous torque/speed gain (Kv) on
device temperature (TD) and a starting friction torque
(To) that was a result of MR fluid sedimentation instatic conditions. Therefore, the relation between
rotational speed and viscous torque can be written as
Tv = To + Kv(TD)v (2)
with v being the rotational speed.
The branch supplied by the current (I) deter-
mines, as previously mentioned, the magnetic con-
tribution (Tm). In the literature, this branch has
been previously modelled as a first-order system [8,
9]. However, experimental results confirmed a Tmgain dependence on the input current. Figure 4
illustrates the steady state Tm versus I characteris-
tics, referring to constant values of both rotational
speed and temperature. The curve was obtained by
subtracting the viscous contribution (torque without
current input) from the measured torque.
The curve depicted in Fig. 4 exhibits a good
agreement with the experimental results presented
in the literature [12, 13]. Moreover, the Tm charac-
teristics should be characterized, as illustrated in
[13], by torque saturation at high current levels. The
absence of such a saturation phenomenon in Fig. 4is an indication that only a part of the full capability
of the device is being used.
The proposed fit for the steady state experimental
results is
Tm = KIr (3)
In order to define a state space model characterized
by the state variable Tm, the steady state equation
(3), with the position
K0
(Tm) = Tr1 =r
m K1=r (4)
can be written as
Tm = K0
(Tm)I (5)
A state space model providing the steady state
response of equation (5) is
_Tm = 1
t(Tm T
r1 =rm K
1=r I) (6)
where t is the time constant.
Equation (6) represents the classic form of state
space non-linear systems that can be generically
written as
_x=Ax+ B x u (7)
where A is the state matrix, B the input matrix, u
the input vector, and x the state vector.
Therefore, the complete system is modelled by
the following set of algebraic-differential equations
Tv = To + Kv(TD)v
_Tm = 1
t(Tm T
r1 =rm K
1=r I)
T= Tv + Tm (8)
with Kv, K, r, To , and t parameters to be identified.
Fig. 3 Logical scheme of MR device
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The described model remains unchanged if the
proposed device as used as a clutch. In this case,
the only difference is in equation (8) in which v
becomes the difference between the input and out-put speeds.
4 PARAMETER IDENTIFICATION
Experiments were performed in order to identify the
values of the model parameters. The measured
quantities were sampled at 10 kHz and 10 s time his-
tories were stored.
Since the parameter Kv depends on temperature,
the identification tests had to be conducted at a
known device temperature (TD) in this case216 1 C. In the remainder of this paper this tem-
perature is called the identification temperature
(Tid). It will then be shown that parameter identifi-
cation at different temperatures is not necessary for
device torque feedback control.
The inputs used in the identification procedure
consisted of a step-up-step-down sequence for the
current and a constant value for the speed. Several
values of both current and speed were considered.
Figure 5 shows a typical current input in which the
presence of an extra current, due to the make, can
be seen. However, it became clear during the experi-ments that this extra current did not affect the sys-
tem output behaviour (Figs 6 and 7).
The identification of parameter values was
achieved by applying a least-square algorithm to
analyse the input and output data. These values
were then used to validate the proposed model.
5 MODEL VALIDATION
The values of the identified parameters (for the case
where TD = Tid) are listed in Table 2. The reportedvalues are the average of five different experimental
tests which provided very stable results. The high
stiffness of the load cell ensured that the identified
value of the time constant (t) could be entirely
attributed to the MR device.Model validation was executed using input time
histories different from the ones adopted during the
identification procedure. Comparisons between the
experimental and simulated data were performed.
Figure 6 shows the current input in a constant rota-
tional speed test (10 rad/s) and Fig. 7 illustrates
model output and measured torque.
Adopting a constant speed input (10 rad/s) and a
quasi-sinusoidal current input (Fig. 8), allowed the
comparison of the evolution of the torque as a func-
tion of time, illustrated in Fig. 9, to be obtained. For
variable inputs in terms of both speed and current(Fig. 10), the obtained results for the evolution of the
torque as a function of time are shown in Fig. 11.
Fig. 6 Current input at a constant rotational speed of10 rad/s
Fig. 5 Typical current input
Fig. 4 Steady state Tm versus Irelationship
Table 2 Identified parameters
Parameter Identified value
t 0.04sKv 0.04NmsK 2.6Nm/Ar 1.6To 0.57Nm
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The following points can be made about these
results.
1. The model response differs very slightly from
the experimental one in the transient after the
step-up of the current input.
2. The agreement between the measured and
simulated results is very good in steady state
conditions.3. The agreement between the measured and
simulated results is also very good in the transi-
ent after the step-down of the current input.
4. Referring to the final seconds of the test illu-
strated in Figs 10 and 11, the goodness of the
model can also be observed with reference to
the descending ramp of the speed input.
5. Peaks can be observed in the measured torque.
These are noise that is created by the inverter in
the non-filtered load signal and, consequently,
they are not reproduced by the model.
Of course, a filtering action on the input and output
signals would lead to much smoother time lines.
Such a filtering action was not performed in this
work in order to prevent any mistakes in the para-
meter identifications created by signal amplitude
and phase distortions.
The influence of temperature is discussed in thenext section.
6 THE INFLUENCE OF TEMPERATURE
As already stated, the parameter identification pro-
cedure was carried out at a device temperature of
216 1 C (Tid). Further tests were performed at dif-
ferent thermal conditions in order to investigate the
influence of temperature. These experimental
results were compared with simulation results. In
the following, the comparisons are made with refer-ence to tests characterized by 1 and 4 A step-up
step-down current inputs. Tests were executed at
device temperatures of 30 and 50 C and a constant
rotational speed of 10 rad/s (Figs 12 to 15).
A difference between simulated and measured
torques can be observed especially in the 1 A tests
(Figs 12 and 13). This difference is caused by the
modified value of the Kv parameter. In fact, the Kvdepends on the fluid viscosity and is consequently
influenced by device temperature (TD). By changing
Kv
to 0.015 Nms, for the 30 C test, and to 0.01 Nms,
for the 50 C test, the results illustrated in Figs 16
and 17 are obtained. They can be considered to be
fully satisfactory for low values of the current.
The updating of the value of Kv is unnecessary for
higher values of the current since at this condition
the contribution of the viscous torque decreases.
Consequently, the influence of the temperature is
less evident for high values of the current. This
observed behaviour confirms the temperature
dependence ofKv and the temperature-independent
nature of all the other parameters.
In the next section it is shown that an incor-rect estimation of Kv can be classified, in a classical
control problem, as a parameter uncertainty that
can be compensated by a torque feedback action.
7 MR FLUID DEVICE CONTROL
The proposed non-linear model was adopted in
order to develop a control system for the trans-
mitted torque.
The control action is constituted by the current
input u. The control system is constituted by amixed scheme (Fig. 18), that consists in the com-
bined action of feedforward and feedback control.
Fig. 7 Comparison of experimental and theoreticalresults on the evolution of the torque as a func-tion of time
Fig. 8 Current input
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The feedforward control action uff is determined by
solving the inverse dynamics of the open-loop con-
trolled system
_x=Ax+ B x uff (9)
where
x= Tm
A = 1t
B x = 1t
(Tr1 =r
m K1=r)
The torque Tm is obtained by subtracting the viscous
torque (Tv) from the target torque. The viscous torque
is estimated by means of the first equation in (8).
A closed-loop control action was introduced
in addition to the open-loop action in order to
compensate for model imperfections and para-
meter uncertainty. The closed-loop controlled sys-
tem equation is
_x =Ax + B x ufb (10)
Fig. 10 Current and rotational speed
Fig. 9 Comparison of experimental and theoretical results on the evolution of the torque as afunction of time
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where x =x xT is the deviation between real anddesired torque and ufb the closed-loop control action.
The feedback control was developed by using the
state-dependent Riccati equation (SDRE). The SDRE
technique is a non-linear control design method for
the direct construction of non-linear feedback
controllers [14]. Using state-dependent coefficient
factorization, system designers can represent the
non-linear equations of motion as linear structures
with state-dependent coefficients. Then, the linear
quadratic regulator technique can be applied to thisstate-dependent state space equation in which all
matrices may depend on the states. The SDRE tech-
nique finds an input u that minimizes the following
performance index
J=1
2
Z0
xTQx + ufbTRufbdt (11)
where ufb = kx and k= R1B(x)TP and P is the
solution of the state-dependent Riccati equation
ATP + PA PB(x)R1B(x)TP + Q = 0 (12)
Fig. 11 Comparison of experimental and theoretical results on the evolution of the torque as afunction of time
Fig. 12 Comparison of experimental and theoretical results for a 1 A step-upstep-downcurrent input and TD = 30 C
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Fig. 13 Comparison of experimental and theoretical results for a 1 A step-upstep-downcurrent input and TD = 50 C
Fig. 15 Comparison of experimental and theoretical results for a 4 A step-upstep-downcurrent input and TD = 50 C
Fig. 14 Comparison of experimental and theoretical results for a 4 A step-upstep-downcurrent input and TD = 30 C
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with Q and R being weight matrices to be matchedexperimentally.
The simulated results on the control perfor-
mances are now discussed and a comparison is
made between feedforward and mixed schemes.
In the simulations the device was assumed to be
initially at rest and then subjected to a ramp-up in
the rotational speed input with a saturation value of
10 rad/s occurring at the instant t = 0.5s.
Figure 19 illustrates the result obtained assuming
that the device temperature is equal to the identifi-
cation temperature (TD = Tid) and considering a step
in the target torque value from 10 to 12 Nm at theinstant t= 3 s .
A mixed control performance is definitely betterduring transient conditions; conversely, the two
control system behaviours can be considered similar
in the steady state regime. Figure 20 indicates the
controlled system response at a new operating tem-
perature TD = 30 C and in the presence of the same
rotational speed input and target torque time line.
This enabled the controlled system performance to
be evaluated in the presence of parameter uncer-
tainty. In fact, in this case, the actual Kv value
(0.015 Nms) is different from the identified one,
used in the feedforward control action.
The steady state feedforward control error isgreater than the one shown in Fig. 19 (TD = Tid). This
Fig. 16 Comparison of experimental and theoretical results for a 1 A step-upstep-downcurrent input, TD = 30 C and Kv= 0.015 Nms
Fig. 17 Comparison of experimental and theoretical results for a 1 A step-upstep-downcurrent input, TD = 50 C and Kv= 0.01Nms
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is because the feedforward control does not use a
modified Kv parameter value to reflect the TD = 30 C
operating temperature. However, the mixed control
system provides a good performance which is dueto the point that the effects created by the change in
the value of Kv are compensated by the intrinsic
robustness of the feedback control. Figure 21 shows
the time line of control action provided by the
mixed control scheme.
A new result was obtained considering an step-uprotational speed input (Fig. 22) and a constant target
torque (10 Nm). The device temperature was
Fig. 18 Control scheme
Fig. 19 Controlled system performance (TD = Tid)
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Fig. 22 Rotational speed
Fig. 21 Control action (u = uff + ufb)
Fig. 20 Controlled system performance (TD = 30 C)
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Fig. 25 Control action (u = uff + ufb)
Fig. 23 Controlled system performance with rotational speed perturbation (TD= Tid)
Fig. 24 Controlled system performance with rotational speed perturbation (TD = 30 C)
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assumed to be equal to the identification tempera-
ture (TD = Tid).
The performances of both the feedforward and
mixed control systems are illustrated in Fig. 23.
Here too, good behaviour can be observed asregards the steady state error for both control sys-
tems. The mixed control system exhibits a good per-
formance in the transient state by minimizing the
effects of a perturbation.
Another simulation was performed at the same
input velocity and target conditions but considered
a different device temperature (TD = 30 C, i.e.
Kv = 0.015 Nms). The obtained results are shown in
Fig. 24. The feedforward control action, based on
the identified Kv value, provides in this case a sub-
stantial steady state error whose value, depending
on the velocity input, becomes significant at higherrotational speeds. Again the mixed control perfor-
mance is fully acceptable. Figure 25 shows the time
line of the control action provided by the mixed
control scheme.
The presented results highlight the effectiveness
of mixed control in the presence of parameter
uncertainty. Use of feedback control combined with
feedforward control, allows the steady state error
and transient delay to be reduced.
8 CONCLUSIONS
A combined theoretical and experimental study has
been performed with the objective of modelling,
parameter identification, and time domain control
design for a MR fluid device.
A model, constituted by an algebraic equation
coupled with a non-linear first-order differential
equation, has been proposed. Model parameters
have been identified by means of experimental tests
carried out on a prototype of the MR fluid-based
deviceComparisons between experimental and simu-
lated data confirmed the soundness of the model.
The state space model, validated in this way, was
used as input to a time domain control design
scheme based on the state-dependent Riccati
equation.
Software simulations confirmed the goodness of
the mixed control scheme for torque regulation.
The described activity highlights the MR fluid
capability to be employed in control aimed devices.
Finally, the influence of temperature on the vis-
cous torque has been discussed in terms of variationin the Kv parameter. The described results show that
the proposed control scheme is able to compensate
for parameter variation effects due to changes in
thermal conditions.
Authors 2011
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APPENDIX
Notation
A state matrix
B input matrix
dg=dt shear rate
H magnetic field
I supplied currentJ functional
k control gain
K magnetic torque/current gain
K0
gain depending on magnetic torque
Kv viscous torque/speed gain
P solution of state-dependent Riccati
equation
Q weight coefficient
R weight coefficient
T transmitted torque
TD device temperature
Tid identification temperature
Tm magnetic torque component
To starting friction torque
Tv viscous torque component
u input vector
ufb closed-loop control action
uff feedforward control action
x state vector
xT target state
x* state error
h fluid viscosity
r exponent in equation (3)
t time constant
ts shear stress
ty yield stress
v rotational speed
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