02 Modelling, Parameter Identification, And Control

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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: [email protected]

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Proc. IMechE Vol. 225 Part I: J. Systems and Control Engineering


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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

550 R Russo and M Terzo



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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

Modelling, parameter identification, and control of a shear mode magnetorheological device 551



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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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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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02 Modelling, Parameter Identification, And Control