Hysteresis Modelling for a MR Damper Jorge De Jesus Lozoya-Santos, S´ ebastien Aubouet, Rub´ en Morales-Men´ endez, Olivier Sename, Ricardo Ramirez-Mendoza, Luc Dugard To cite this version: Jorge De Jesus Lozoya-Santos, S´ ebastien Aubouet, Rub´ en Morales-Men´ endez, Olivier Sename, Ricardo Ramirez-Mendoza, et al.. Hysteresis Modelling for a MR Damper. 7th EUROSIM Congress on Modelling and Simulation (EUROSIM 2010), Sep 2010, Prague, Czech Republic. pp.71. <hal-00504923> HAL Id: hal-00504923 https://hal.archives-ouvertes.fr/hal-00504923 Submitted on 21 Oct 2010 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.
Transcript
Hysteresis Modelling for a MR Damper
Jorge De Jesus Lozoya-Santos, Sebastien Aubouet, Ruben Morales-Menendez,
Jorge De Jesus Lozoya-Santos, Sebastien Aubouet, Ruben Morales-Menendez, Olivier Sename,Ricardo Ramirez-Mendoza, et al.. Hysteresis Modelling for a MR Damper. 7th EUROSIMCongress on Modelling and Simulation (EUROSIM 2010), Sep 2010, Prague, Czech Republic.pp.71. <hal-00504923>
HAL Id: hal-00504923
https://hal.archives-ouvertes.fr/hal-00504923
Submitted on 21 Oct 2010
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.
An experimental dataset of a commercial Magneto-Rheological (MR) damper is exploited foridentification of a Hysteresis-based Control-Oriented model. The model wellness for hystere-sis, saturation and transient responses is shown through validation with experimental data. Astudy case that includes a Quarter of Vehicle (QoV) shows that the hysteresis phenomena couldaffect the primary ride and vehicle handling. Several analysis based on open and closed loopsimulation demonstrated that hysteresis must be considered for controller design.
Keywords: hysteresis, MR damper model, model simulation, vehicle dynamics
Presenting Author’s BiographyRuben Morales-Menendez holds a PhD Degree in Artificial Intelligencefrom Tecnologico de Monterrey. From 2000 to 2003, he was a visitingscholar with the Laboratory of Computational Intelligence at the Univer-sity of British Columbia, Canada. For more than 23 years, he has been aconsultant specializing in the analysis and design of automatic control sys-tems for continuous processes. He is a member of the National ResearchersSystem of Mexico (Level I) and a member of IFAC TC 9.3.
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1 IntroductionThe Magneto-Rheological (MR) damper is a non-linearcomponent with dissipative capability used in the con-trol of semi-active suspensions, where the damping co-efficient varies according to the applied electric current.The modelling of the force-velocity curve is not a triv-ial task due to the hysteresis loop and nonlinear behav-ior. The MR damper inherits the hysteresis and bilin-ear behavior from its MonoTube (MT) mechanical de-sign. Several MT damper modelling approaches showthat the acceleration zdef and velocity zdef as inputvariables are needed for a precise hysteresis simulation,[1, 2], and for a realistic simulation of chassis accel-eration, suspension deflection and road holding, [3, 4].A MR damper model including the hysteresis becomesnecessary in order to obtain precise simulations.
This paper presents a MR damper model based on thearrangement of two devices that models the passiveforce consisting of a spring and a damper, and a damperwhich models the force due to the current effect over theoil viscosity. The model captures the hysteresis and thecurrent effect achieving frequency independent param-eters. The study of the hysteresis compliance is donethrough a comparison of two modelling approaches inthe suspension of a Quarter of Vehicle (QoV).
This document is organized as follows. A theoreticalbackground on MR damper modelling and its challeng-ing issues are exposed in section 2. Section 3 presentsthe proposed model, and experimentation and identifi-cation procedure. A study case is presented in Section4. Section 5 discusses the findings and results. Theconclusion is presented in section 6. Table 1 defines thepaper nomenclature.
2 Theoretical backgroundSome fundamental concepts are reviewed, [5]:
(a) When a certain magnetic field is applied to the MRfluids, the particles in the fluids are polarized and theyform polarization chains, in the parallel direction to theapplied field.
(b) When the MR fluids are subjected to an externalshear stress in the perpendicular direction to the mag-netic field, those polarization chains can resist the shearstress to some extent, and the MR fluids behave in a vis-coelastic way. This region is referred to as the pre-yieldregion. When the external shear stress is increased andexceeds a certain value, the polarization chains will bebroken and MR fluids becomes in regular Newtonianfluids; this region is referred to as the post-yield region.
(c) The gradually decreasing shear stress links up thepolarization chains, but the stress value to do the link isless than before the polarization chains break happens,producing the hysteresis. The switch of MR fluids fromviscoelastic behavior to regular Newtonian fluid behav-ior is referred to as yield, and the value of the externalshear stress at this point is thus known as yield stress,and depends on the intensity of the applied field. Sincethe polarization chains are increased with the magnetic
Tab. 1 Nomenclatureai Coefficient depending
polynomially on the currentcMR MR damping coefficient in (N ·A)/mcp Passive damping coefficient in Ns/mhh Coefficients dealt with the horizontal
hysteresiskp Passive stiffness coefficient in N/m
fMR MR damping observed force in Nfp Force due to damper mechanics (N)fI Force due to MR fluid (N) and I
fMR Predicted forceh Input magnetization rate into a
ferromagnetic materialc2, c3 Magnetic saturation due to
remanent magnetism and minor loopsf+ Positive magnetic saturation level
for positive hf− Negative magnetic saturation level
for negative hhi Coefficient for the minor loops
and dc biashs1, hs2, hs3 Coefficients for the hysteresis
hv Coefficient for vertical hysteresisI Applied current in Aq Exponent for I in the polynomialr Polynomial order for the polynomial
current dependencyt Time
y1, y2 Coefficients for the MRsigmoid behavior
x, z, zdef Piston deflection in metersx, z, zdef Piston deflection velocity in m/s
x, z Piston deflection acceleration in m/s2
zs Chassis velocity in m/szs Chassis acceleration in m/s2
w Disturbation shaped as chirp sinusoidalαq,i i-th coefficient for the polynomial
current equationθ Parameters vector
ϕdataset the input dataset matrixi ith input dataset row
Zdef , Zdef , Column vectors of theexperimental dataset
Zdef , I experimental datasetω Frequency in rad/s
fXX−MR MR damping force estimation.The letters XX can be CO or HCO
fsteering Force generated for a steering actionf : a 7→ b The function f maps the element a to
field, the shear stress need to be large in order to breakthem. The larger the magnetic field, the larger the yieldforce.
A MR damper is usually characterized by the displace-ment and/or velocity of the piston, the electric currentapplied to the coil as inputs and the force generated onthe piston as output. A static MR damper model is buildhaving direct or instantaneous links between input vari-ables, thus it esimates the force using the present cur-rent and displacement/velocity and not earlier values. Adynamic MR damper model is also built with the samevariables but these variables change before affecting theforce value by their intrinsic dynamics, and their valueswill thereby also depend on early current or displace-ments.
A classification for this device is proposed by its orig-inal structure as: passive and I-driven. A passive orig-inal model does not include the electric current as in-put variable while the I-driven model does. In the lit-erature, the MR damper modelling has been based inall the known model structures: phenomenological (pa-rameters which meaning is related to the mechanicalparts, physical meaning), semi-phenomenological (pa-rameters with physical meaning) and the black-box sup-port.
2.1 Passive models
The MR damper is often described by the general form:
fp = [ g(x, x) | θ ]
where the fluid is MR. Therefore, the study of the staticcurves force-velocity and the transient response havebeen the basis of such models. The consequence is themissing of the electric current (or voltage) as a modelinput, Fig. 1.
MR Damper
ModelStatic or dynamic
x
dx/d t
(I)
fMR
θ
Parameters
other
Fig. 1 Passive MR damper model
The electric current effect on the damping force is ad-dressed by the representation of each model parame-ter as a polynomial function of the current; hence, themodel is represented as
fMR = [ g(x, x) | θ(I) ] ai(I) =∑r
q=0 αq,i · Iq (1)
Consequently, there is an over-parameterization, andif the polynomial order is greater than 1, the nonlin-ear complexity is augmented. The main properties ofthe model are not improved. There are several impor-tant works: Bingham model [6], Bouc Wen Modified[7], Polynomial Approach [8], Three Parameters model
Tab. 2 Classification of the literature passive models.Application capability of the model: d for diagnostic, efor estimation and c for controller synthesis. Hystere-sis means the model capability in order to capture thehysteresis. θ length is the model number quantity. θ(I)length defines the possible number of parameters whenthe model must be polynomially dependent on the cur-rent.
Features Bouc Poly- Semi- PhaseWen nomial physical transition
ModifiedApplication e e d dθ length 7 12 5 5
Hysteresis Yes Yes No NoI as input No Yes No Noθ(I) length 14 - 30 30
Type Dyn Static Static DynFrequency Yes No Yes Yesdependentparameters
[9], Sigmoidal-based Behavior model [10], and Phase-Transition model [5]. Table 2 compares some works inthis field.
2.2 I-driven models
The MR damper is described by
fMR = [ g(x, x, I) | θ ]
This model considers the applied current as a model in-put variable, Fig. 2.
MR Damper
ModelStatic or dynamic
x
dx/d t
I
fMR
θ
Parameters
other
Fig. 2 I-driven MR damper model
The parameter identification requires of experimentsobtained with persistent excitation in both displacementand current. The modelling contributions are classi-fied as Surface Response Method or Statistical model[11], the Nonlinear Auto-Regressive with eXogenousinput (NARX) model [12], NARX based in Neural Net-works (NNARX) [13], considered as black box or non-parametric models, and the Dynamic Transfer functionCurrent-Force [14], which analyzes the transient behav-ior of the force due to the electric current. Table 3 com-pares some features of those approaches.
Tables 2 and 3 compare the main features and character-istics of MR damper models. Based on this comparison,some demands are found: (a) the hysteresis phenomenahas not been addressed with success given that the ana-lytical models are based on static force-velocity curveswhere the applied electric current is constant and theylack of the acceleration z as input, an important iner-tial element in the damping force [1], (b) If the experi-mental database has static currents, the current could bedefined as a varying parameter, where the continuous-time variation of this parameter is considered, (c) thereis a necessity for a control-oriented model given that theliterature models are proper for estimation or diagnosisand their application for controller synthesis becomesinvolved.
3 MR damper modellingA general MR damper structure model can be describedby two dampers in parallel: a damper with constantshear stress (passive) and a damper with variable-shearstress (semi-active) due to the variation of the appliedelectric current. The sum of these components yieldsthe MR total damping force fMR.
fMR = fp(z, z) + fI(z, z, I)
fp = kpz + cpz + o(z, z, z)
fI = cMR · I · g(z, z) (2)
where o(·) and g(·) are nonlinear functions. o(·) de-scribes hysteresis and g(·) describes the semi-active be-havior.
3.1 Hysteresis-based Control-Oriented (HCO)Model
Equation (3) can represent a magnetic saturation curvewith sigmoid T-shape T for a h, [15].
T (h) = A0 · h+B0 · tanh [C0 · h] (3)
In order to develop the mathematical expression for ahysteresis loop, the translation of the T (h) functionhorizontally is done by ±a0 as vertically by ±b1. Thesign ± specifies directions,
where the measure of the horizontal shift a0 representsthe value of the coercive force, where the hysteresisloop intersects the horizontal axis. The value of b1 isdetermined by hmax, the maximum value of h wherethe two curves f+ and f− intersect. The values of c2and c3 represent the minor loops and the remanent mag-netism. The minor loops, is a collective name of all theloops, which have at least one extreme different fromthe major loop extreme. When the exciting magneticfield is interrupted during the process of magnetization,then the magnetization of the sample will not stay thesame, it declines to a value below the value determinedby the field at the point of the interruption on the hys-teresis loop, phenomenon called remanent magnetism.
The Force-Velocity (FV) damper map is a well knowndiagram with high complexity. It exposes the dampingratio ζ behavior. If ζ has a one-to-one relation, the nextsentence is true, based on (3).
Let ζ : z 7→ h, fMR 7→ T (h) be defined byζ(z, fMR) = fMR/z.
represents the damper behavior without hysteresis andconstant current. In order to obtain a more precisemathematical approach of a damper, the following as-sumptions and analogies are proposed.
By assuming a harmonic motion of the damper piston,let z, z, z approximated by:
z ∼ R · sin(ω · t) (6)z ∼ ω ·R · cos(ω · t) (7)
z ∼ −(ω)2 ·R · sin(ω · t) (8)
It has been shown that the acceleration deflection z isthe stronger influence on shaping of the hysteresis, [1,2]. The zdef inertial effect contributes to the hysteresisoffset. From equations (4, 5), and based on [1, 2, 10,16], the static mapping defined in [15] is extended to adynamic mapping
Hh : hh · z 7→ a0dynamic (9)Hi : hi · z 7→ c1,3dynamic (10)
Hv : hv · z 7→ b1dynamic (11)
Thus, the values for the coefficients defined in (4) and(5) will change according to z, z and z. The hystere-sis loop will be shaped depending on the piston motiondynamics. The proposed model is defined by:
fMR = kpz + cpz + hv · z +hs2 · tanh (hs3 · z + hh · z) +cMR · I · tanh (y1 · z + y2 · z) (12)
where cp = hs1 and kp = hi, and the sigmoid functiondefines the shape accuracy. This model is an I-drivenfull model because it includes the current, the deflectionand its first and second derivatives of the deflection asinputs.
. Modifying the Semi-Phenomenological model [10],an I-driven model (13) can be proposed. This modellingapproach performs good simulations of the bi-viscousand saturation behavior but limits the hysteresis repre-sentation.
A Frequency modulated displacement with ICPS elec-tric current as input signal was exploited in order toidentify the MR damper in a experimental system, [16].The models represented by equations (12,13) are iden-tified by using the nonlinear least square curve fit algo-rithm with the following cost function:
minθ
||fMR(θ, ϕdataset)− fMR||22 =
minθ
∑i
(fMR(θ, ϕi)− fMRi
)where ϕdataset = [Zdef , Zdef , Zdef , I].
4 Study caseThe system consists of a simple model of a Quarter ofVehicle (QoV). The main components are the sprungmass (ms) and the unsprung mass (mus). The springwith a stiffness coefficient ks and a semi-active damperbuilt the suspension between masses. The stiffness co-efficient kt models the wheel tire. The vertical positionof the mass ms (mus) is defined by zs (zus). It is as-sumed that the wheel contact is kept.
The used QoV parameters corresponds to a RenaultMegane CoupeTM model (see [17]) whose values are:ms = 315 Kg, mus = 37.5 Kg, ks = 29500 N/m,kt = 210000 N/m. The damper parameters are definedaccording to equations. (13) and (12).
Two QoV simulation systems with a semiactive suspen-sion were implemented. They used MR damper models(12) and (13) resulting equations (15) and (17), calledlinear QoV and nonlinear QoV respectively. The steer-ing force is considered zero. The linear word is used inthe sense of the hysteresis phenomena presence on theMR damper.
The dynamical equations for the linear QoV are gov-erned by:
mszs = −ks · z − fCO−MR + fsteeringmuszus = ks · z + fCO−MR − kt (zus − zr)
(14)where the fCO−MR is estimated by:
fCO−MR ≈ cp · z + kp · z+cMR · I · tanh (y1 · z + y2 · z)
(15)
and it uses the parameters for the CO MR damper modelshown in Table 4.
The dynamical equations for the nonlinear QoV aregoverned by:
mszs = −ks · z − fHCO−MR + fsteeringmuszus = ks · z + fHCO−MR − kt (zus − zr)
(16)where the fHCO−MR is estimated by:
fHCO−MR ≈ kpz + cpz + hv · z +hs2 · tanh (hs3 · z + hh · z) +cMR · I · tanh (y1 · z + y2 · z)
(17)
and it uses the parameters for the HCO MR dampermodel shown in Table 4.
It is worth to note that when the applied current equals0 A, the passive damping is approached by a linear co-efficient in equation (15). The frequency response orpseudo-bode, [18]), of both systems exposes the con-trollability margins with and without hysteresis inclu-sion.
Two simulation tests were done: open and closed con-trol systems, Figure 3.
Road (m)
Input(Am/s)
z
z -
s
s
z us
SA-QoV
Chirp signal
0-20 Hz
with 15mm
peak
z’suu f
(z - z) s us
Power Driven
Damper
(Morselli, 2008)
Controller
Filterfrequency = 20 Hz
Fig. 3 Diagram of simulation for the quarter of vehiclein open and closed loop. The controller input signalsare considered measurable. The filter block is requiredin order to assure soft changes when the control lawswitches between the commanded current values, de-fined in (18). The filter cut-off frequency must be atleast 20 hz in order to cover the automotive bandwidth.
The applied current was {0.001,0.5,1,1.5,2,2.5} Afor the open control system. For the closed con-trol system, comfort oriented controller Power-Driven-Damper Control (PDD), [19], evaluates both systems.The specifications for the comfort, [20], are defined inthe span of [0-10] Hz, the maximum gain of the fre-quency response zs/zr must be kept low below 200.The (PDD) control strategy is obtained from control ofHamiltonian port systems and claims for soft manipu-lations and good performance under ω = 120 rad/s.
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The closed control input input is the current, hence, thecontrol strategy has been modified in order to be properfor the MR QoV suspension. The modified PDD controllaw is:
Imin
if K · zdef · zdef + bmaxz2def < 0
Imax
if K · zdef · zdef + bminz2def ≥ 0
(Imax + Imin)/2if zdef = 0 and zdef = 0
Imax · zdef/zdef otherwise(18)
where K = k + kp, bmax corresponds to the MRdamper model force evaluated in zdef at maximum cur-rent 2.5 A and divided between zdef m/s, bmin corre-sponds to the MR damper model force evaluated in zdefat cero current and divided between zdef m/s. The val-ues for Imin and Imax where defined as 0.2 and 2.5 A.See Figure 3 for the simulation diagram and filter fre-quencies for the controller.
5 DiscussionTable 4 shows the identified parameters.
The descriptive force-velocity curves shows the goodtracking of the hysteresis with the model HCO, left topplot in Fig. 4. The model CO makes good saturation es-timation but at low velocity the precision is small, righttop plot in Fig. 4. The force-velocity curves, bottomplot in Fig. 4, exposes the covering done by each modelversus real data where the real hysteresis matches in amore precise shape for model HCO.
In transient response, the model HCO becomes moreaccurate for small and big forces. The model CO failsthe estimation for small forces (≤ 200 N) because theinertial effects are missing the model. See the modelserror at low forces on Fig. 5.
The classification of the evaluated models shows bet-ter features than the aforementioned reviewed, Table 5,adding an easier process for their identification. A wideanalysis is necessary in order to define the frequencydependence of the parameters.
The open loop tests exposes a different controllabilitywhen the suspension includes hysteresis and when itdoes not, see Fig. 6. In the span of primary ride (0-2.5Hz) the nonlinear case shows a more realistic and con-strained controllability than its linear counterpart. Formaximum current and rattle-space frequency (the chas-sis and the tire fixture can shock each other, approx.
−1
0
10.5
−0.5
force(=)kN
HCOmodel
Realdata
COmodel
Realdata
−1
0
10.5
−0.5
force(=)kN
−1
0
10.5
−0.5
force(=)kN
−0.4 −0.2 0 0.2 0.4
velocity(=)m/s
HCOmodel
Real data
COmodel
Fig. 4 Attitude of MR damper models in force-velocitycurves: detailed and general views.
Tab. 5 Classification of the proposed models. The cate-gories are the same as for table 2.
1.8 Hz), referring to the linear case, the zdef showsvery low gain and very good comfort while the non-linear case is more conservative. In the spans of (2.5-11Hz) and (16-20 Hz) both approaches offer similar per-formances, but the minimum gain is greater with hys-teresis. In (11-16 Hz), the simulation results show howthe tire-hop (the tire can be separated from the surface)frequency is affected by the hysteresis being an interest-ing and important result in the evaluation on controllers,given that this frequency is key for the vehicle handling.The simulation allows to infere that in this bandwidth,high currents are desired in order to improve the vehiclestability.
The closed loop test offers more illustrative results. Thecontroller is very efficient when the semi-active suspen-sion does not include the hysteresis (similar to that re-ported in [19]). When the hysteresis is taken into ac-count, i. e. the MR damper is HCO model, the con-troller performs as expected for 2-8 Hz but for the rattle-space frequency the performance is very poor from theobtained without hysteresis.
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1.04 1.05 1.06 1.07 1.08
x 104
−1000
0
1000
samples
forc
e(=
)N
HCO Model
CO Model
Real data
Fig. 5 MR damper models transient response
2 4 6 8 10 12 14 16 18 200
500
1000
frequency [Hz]
0
200
400
600
800
1000
1200
ga
in
Pseudo Bode of zs’’/z
r
0.001 A
2.5 A
ga
in
Fig. 6 Performance of semi-active suspensions, with and without hysteresis, in order to modify the control objective(comfort) in a semi-active suspension in the span of 0-2.5 A of current. Top plot is the linear QoV case, bottomplot is the nonlinear QoV case.
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2 4 6 8 10 12 14 16 18 200
100
200
300
400
500
600
700
800
900
1000
1100
frequency [Hz]
ga
in
Pseudo Bode of zs’’/zr
PDD with HCO model
PDD with CO model
Fig. 7 Performance of the semi-active suspension, with and without hysteresis, in closed loop for the PDD con-troller.
6 ConclusionThis work proposes a MR damper model called (HCO)with several features: a simple structure, standard iden-tification procedure, precision in hysteresis simulation,and proper structure for controller synthesis. This mod-elling approach is compared with a model based ontransient response. A study case, a Quarter of Vehiclewith a MR damper shows that the hysteresis could affectthe primary ride and vehicle handling. Missing the hys-teresis phenomenon decreases the negative effects onthe rattle-space and tire-hop frequencies. Hence, thiswork focuses the attention to more accurate specifica-tion in the simulation of vehicle behavior for controllerevaluation purposes. The example was illustrative andrequires more analysis.
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