Modeling, Identification and Control, Vol. 36, No. 3, 2015, pp. 133–142, ISSN 1890–1328

On Implementation of the Preisach ModelIdentification and Inversion for Hysteresis

Compensation

Jon Age Stakvik 1 Michael R.P. Ragazzon 1 Arnfinn A. Eielsen 1 J. Tommy Gravdahl 1

1Department of Engineering Cybernetics, Norwegian University of Science and Technology, N-7491 Trondheim,Norway. E-mail: jas@kelda.no, {ragazzon,eielsen,jan.tommy.gravdahl}@itk.ntnu.no

Abstract

A challenge for precise positioning in nanopositioning using smart materials is hysteresis, limiting posi-tioning accuracy. The Preisach model, based on the delayed relay operator for hysteresis modelling, isintroduced. The model is identified from experimental data with an input function ensuring informationfor all input levels. This paper presents implementational issues with respect to hysteresis compensationusing the Preisach model, showing the procedure to follow, avoiding pitfalls in both identification andinversion. Issues due to the discrete nature of the Preisach model are discussed, and a specific linearinterpolation method is tested experimentally, showing effective avoidance of excitation of vibrationaldynamics in the smart material. Experimental results of hysteresis compensation are presented, show-ing an approximate error of 5% between the reference and measured displacement. Consequences of aninsufficient discretization level and a high frequency reference signal are illustrated, showing significantdeterioration of the hysteresis compensation performance.

Keywords: Hysteresis, Preisach, Identification, Inversion

1 INTRODUCTION

Materials exhibiting both sensing and actuation capa-bilities, arising from a coupling of mechanical proper-ties with applied electromagnetic fields, are commonlyreferred to as smart materials Moheimani and Goodwin(2001). These materials include, for instance, piezo-electric materials and shape memory alloys, which arematerials that show nonlinear hysteretic behavior Tanand Baras (2005). Hysteresis causes the output to lagbehind the input, and is the main form of nonlinearityin piezoelectric materials Devasia et al. (2007). Withan accurate description of hysteresis, the effect can becompensated for Eielsen et al. (2012); Iyer and Tan(2009); Leang and Devasia (2006); Croft et al. (2001).

Hysteresis models can be roughly classified into twogroups, physical-based and phenomenological-based

Tan and Baras (2004); Esbrook et al. (2013). TheJiles-Atherton model of ferromagnetic hysteresis Jilesand Atherton (1986), is an example of a model basedon a physical description that can describe hysteresisRosenbaum et al. (2010). However the derivation ofsuch physical models can be an arduous task, and of-ten result in high order models not suited for practi-cal applications Ismail et al. (2009). Phenomenologicalmodels, on the other hand, aim only to approximatethe physics, thus giving simpler models. The Preisachoperator model Iyer and Tan (2009); Hughes and Wen(1997); Iyer and Shirley (2004); Tan et al. (2001); Tanand Baras (2005); Leang and Devasia (2006); Zhao andTan (2006) for hysteresis modelling was first regardedas a physical model for hysteresis based on some hy-pothesis concerning the physical mechanisms of mag-netization Mayergoyz (2003). However in later years

doi:10.4173/mic.2015.3.1 c© 2015 Norwegian Society of Automatic Control

Modeling, Identification and Control

u

y

β α

Figure 1: Delayed relay operator.

InverseOperator

HysteresisOperator

ur y u

Figure 2: Illustration of feed forward hysteresis com-pensation.

the model has been recognized as a phenomenological-based model mainly due to a more general description.

The idea of Preisach models, including the Preisachoperator, the Krasnosel’skii-Pokrovskii (KP) operatorRiccardi et al. (2012); Tan and Bennani (2008) andthe Prandtl-Ishlinskii (PI) operator Al Janaideh et al.(2011); Macki et al. (1993) is to calculate an output asa sum of weighted basis operators. The Preisach oper-ator is described as a delayed relay element, depictedin Fig. 1, the KP operator as a delayed relay elementwith final slope and the PI operator as play and stopelements. Consequently, due to their similar structure,these operators are referred to as Preisach-type oper-ators Tan and Bennani (2008); Iyer and Tan (2009);Rosenbaum et al. (2010). This paper will focus specif-ically on the Preisach operator, for more informationabout the PI and KP operators the reader is advisedto consult Tan and Bennani (2008) as a starting point.

Numerous papers have presented hysteresis com-pensation by employing the Preisach model, however,there are few references to the difficulties and chal-lenges with the implementation of such a hysteresiscompensation scheme. This paper demonstrates im-portant aspects to keep in mind when implementingan identification and inversion scheme based on thePreisach operator. The Preisach operator has beenchosen due to its popularity, and its common use inthe literature. Several implementational issues arediscussed and explained with respect to the Preisachmodel, showing the procedure to follow for hysteresiscompensation, avoiding pitfalls in both identificationand inversion. In Tan et al. (2001) interpolation is

proposed to avoid discrete inputs from the inversionalgorithm, however no method for choosing interpola-tion points was proposed. This paper experimentallytests a specific linear interpolation method, showing ef-fective avoidance of vibrational dynamics in the smartmaterial.

In order to compensate for hysteresis with thePreisach model, the Preisach density function, whichcorresponds to a weight for each delayed relay element,must be identified. One of the first methods proposedfor identification was to twice differentiate the Everettfunction, obtained by applying first order reversal in-puts to the material Mayergoyz (2003), i.e. changingthe input signal to the opposite direction. Howeverthis method involves differentiation of a measurement,and will consequently be highly sensitive to measure-ment noise. The most common method for identifica-tion of the Preisach density function is based on theconstrained least squares method Iyer and Tan (2009);Iyer and Shirley (2004); Galinaitis et al. (2001); Tan(2002); Eielsen (2012). In contrast to linear systems,hysteresis is a rate-independent effect, i.e. that in or-der to maximize the information through identification,only the input magnitude needs to be varied, and notthe input frequencies Iyer and Shirley (2004).

Compensation of hysteresis is commonly achievedby feed forward of an inverse hysteresis input Deva-sia et al. (2007), illustrated in Fig. 2. This has, forinstance, been done for the Coleman-Hodgdon modelEielsen et al. (2012) and the Preisach model with theconstruction of a right inverse of the hysteresis modelTan and Baras (2005); Hughes and Wen (1997); Croftet al. (2001). However analytical solutions of Preisach-type operators generally do not exist, with the excep-tion of the PI model described with play operators,which inverse turns out to be a stop-type PI opera-tor Kuhnen (2003); Tan and Bennani (2008). Conse-quently, the inversion of the Preisach operator oftenhas to be carried out iteratively Iyer et al. (2005). Inthis paper this is done based on a closest match algo-rithm Tan et al. (2001).

It is well known that negative phase introduces aphase lag between the input and output, which mustnot be confused with the lagging behind property ofthe hysteresis nonlinearity Devasia et al. (2007). Dueto this, if a time delay and/or a low pass filter ispresent, the system will experience a phase lag in ad-dition to the real hysteresis. However such a model isonly valid for frequencies close in value to the frequen-cies used to identify it Stakvik (2014). A time delay ofTd = 4.58× 10−4s has, for instance, been reported in acommercial available atomic force microscopy (AFM),where the source of this time delay is speculated to bethe displacement sensor in the AFM Ragazzon et al.

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(2014).The remainder of this paper is organized as follows.

The Preisach model is described in Section 2. Section3 first states the experimental setup, before the identi-fication procedure is explained together with some im-plementational aspects. In Section 4 experimental re-sults of hysteresis compensation are presented, study-ing both the effect of low discretization and high fre-quency on the reference signal. Finally, in Section 5some concluding remarks are drawn.

2 MODELING

2.1 The Preisach Operator

The Preisach operator is built up by the delayed relayoperator, shown in Fig. 1, where the output y to aninput u, for a pair of thresholds (β, α), with β ≤ α isgiven as

y(t) = Rβ,α[u, ξ](t) =

1, for u(t) > α

y(t− 1), for β < u(t) < α

−1, for u(t) < β

(1)where t is the time, y(t) is the relay output at time t,defined from the previous output y(t − 1) = ξ whereξ is the state of the relay, and Rβ,α the output of therelay corresponding to the (β, α) pair.

Further, the output of the Preisach operator is cal-culated as a weighted superposition of delayed relays.For an applied input uβ,α ∈ {+1,−1}, the output ofthe Preisach operator is given by Mayergoyz (2003)

y(t) =

∫∫α≥β

µ(β, α)Rβ,α[u, ξ](t)dβdα (2)

where µ(β, α), called the Preisach density function, is anonnegative weight function, representing the weightsof each hysteron in the Preisach plane S = {(β, α) :α ≥ β, α ≤ αm, β ≥ βm}, where αm and βm refer tothe highest and lowest values for α and β respectively.The Preisach plane S can be geometrically divided intotwo subregions, S+, which corresponds to relays withoutput value +1, and S−, corresponding to output val-ues of −1.

To understand the changing states of the Preisachplane, assume that at some initial time t0, the inputu(t0) = u0 < βm. Hence, all delayed relay elementshave output value −1. Further, assume that the in-put is increased monotonically until some maximum attime t1, where the input value is u1. The correspond-ing Preisach plane is illustrated in Fig. 3a, where theboundary between S+ and S− is the horizontal line

β

αα = β

βm

αmS−

S+

u1

(a)

β

αα = β

βm

αmS−

S+

u2

(b)

Figure 3: Illustration of the staircase memory curve ofthe Preisach Plane.

α = u1. Then all elements below the memory curveare turned on, that is, they have a α value lower thanu1, likewise, all hysterons with α > u1 are turned off.Next, the input is decreased monotonically until someinput u2 at time t2, then all hysterons with a β valuehigher than u2 will be switched off. This results in thestaircase memory curve shown in Fig. 3b.

Based on the regions S+ and S− the output in (2)can be rewritten to

y(t) =

∫∫S+

µ(β, α)dβdα−∫∫S−

µ(β, α)dβdα (3)

where the negative contribution is subtracted from thepositive contribution. Based on this equation the rate-independent property of the Preisach operator is ob-served. The output of (3) only depends on the regionsS+ and S−, which again is defined from the past se-quence of local maxima and minima of the input. Con-sequently, since the output only depends on the levelsof the input, and not the frequency, the Preisach oper-ator is rate-independent.

2.2 Discretization

In order to implement the discrete Preisach modelnumerically, the Preisach plane must be discretized.A common method of doing this is to partition thePreisach plane into subregions, shown in Fig. 4, wherea discretization level of nh = 4 results in nh + 1 inputlevels. The weight µ(β, α) is then described as a centerweight, shown as red dots in the same figure. This im-plies that if the red dot is a part of the S− region, thecontribution from this subregion will be negative, cor-respondingly, if the dot is part of S+ the contributionwill be positive. The input levels n = 1 and n = h+ 1refer to βm and αm respectively, which define the rangeof the Preisach model as shown in Fig. 3.

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u5

u4

u3

u2

u1

u1 u2u4 u5 β

α

Figure 4: Partition of Preisach plane with centerweighting masses.

Figure 5: Nanopositioning stage.

3 IDENTIFICATION

3.1 Experimental setup

The aim of this work is to identify and compensatefor hysteresis by running tests on a piezoelectric actu-ator in a nanopositioning laboratory. The experimen-tal setup consists of a dSPACE DS1103 hardware-in-the-loop system, an ADE 6810 capacitive gauge, anADE 6501 capacitive probe from ADE Technologies,a Piezodrive PDL200 voltage amplifier, the custom-made long-range serial-kinematic nano-positioner fromEasyLab (see Fig. 5), and two SIM 965 programmablefilters. The capacitive measurement has a sensitivityof 1/5 V/µm and the voltage amplifier has a gain of20 V/V. The programmable filters were used as recon-struction and anti-aliasing filters. With the DS1103system, a sampling frequency of 100 kHz is used in allexperiments.

3.2 Constrained Linear Least SquaresIdentification

In this paper, all identification schemes were conductedoff-line with a constrained linear least squares method.Recall that the Preisach density function, µ(β, α) isnonnegative, which gives the constraints for the least

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Figure 6: Fig. (a) shows an example of an input sig-nal with sufficient reversals for identificationof a Prisach model with discretization levelnh = 15. Fig. (b) shows the correspond-ing Preisach distribution based on the inputsignal in Fig. (a).

squares estimate µ(β, α) ≥ 0, where µ(β, α) is the esti-mate of µ(β, α). In order to estimate µ(β, α), a time se-ries of measured input and outputs were used to createan over-determined system of linear equations basedon a discretized form of (2)

y(t1)y(t2)

...y(tnt)

=

A1R1(t1) · · · AqRq(t1)A1R1(t2) · · · AqRq(t2)

.... . .

...A1R1(tnt) · · · AqRq(tnt)

µ1

µ2

...µnq

+η0

(4)where the vector consisting of y values is the measuredhysteretic output, η0 is a constant contribution, Ri(tj)is the output calculation for Preisach element i at timej, corresponding to Rβ,α[u, ξ](ti) in (2), nt is the num-ber of samples and nq is the number of Preisach ele-ments, given as

nq =nh(nh + 1)

2(5)

where nh is the level of discretization.The parameters to be identified, µ and η0, which

is the estimate of η0, is then found using the pseudoinverse as

µ = A+(y − η0) (6)

where η0 is a measurement bias component, A+ is thepseudo inverse of the matrix A in (4), µ is the vectorof estimated Preisach weights and y is the measuredoutput of the hysteresis. This identification schemecan be performed by the procedure outlined in the nextsubsection.

When identifying the Preisach density function, theinput used for identification is of vital importance.Choosing an input with an equal or larger number of

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(c) Large input amplitude, [-150,150].

Figure 7: Showing the detrimental effects on the Prisach weight function using wrong input amplitudes foridentification of the discrete model. The model is defined for an input range of [-90,90].

input reversals compared to the number of discretiza-tion levels, nh, results in at least one reversal for eachPreisach operator. Consequently, such a number of in-put reversals will guarantee that every delayed relayelement is switched on and off at least once, provid-ing excitation of all Preisach elements. Such an inputis shown in Fig. 6a, where a sinusoidal signal withincreasing amplitude provides a sufficient number ofinput reversals for identifying a Preisach model withnh = 15. An example of an identified Preisach distri-bution, obtained from the experimental setup, can beseen in Fig. 6b.

Since the hysteresis nonlinearity is rate-independent,the frequency of the input signal does not affect thehysteresis response. However due to the dynamics inthe smart material and low pass filters in the exper-imental setup, the frequency of the identification sig-nal must be restricted. This is the case in piezoelec-tric actuators, where a creep effect appears for lowfrequencies, while vibration dynamics are excited athigh frequencies. In addition, low pass filters havean increasing negative phase for increasing frequencies.For the experimental setup applied in this work, thecreep effect occurs below 5 Hz, while the vibration dy-namics have a resonance frequency of about 780 HzStakvik (2014). To avoid vibrations, the fundamen-tal frequency of the input should be at least a factor often below the resonance frequency, preferably even lessDevasia et al. (2007). Consequently, the identificationsignal should be chosen to avoid both high and lowfrequency issues. A more detailed discussion of theseissues are presented in Croft et al. (2001), where theeffect of both creep and vibration are modelled priorto the hysteresis identification.

3.3 Aspects of Implementation

The estimated bulk contribution, η0, in eq. (6) can beviewed as a constant bias component outside the range

of the Preisach model. To identify both µ and η0, arepeating two step procedure has been applied. Firstly,the Preisach weight µ is identified with an initial η0,typically zero, satisfying the constraints. Secondly, thebulk contribution η0 is estimated based on the firstestimate of µ. This procedure is repeated until the es-timate of η0 converges to some value. In this paper theMATLAB functions lsqnonneg and lsqnonlin are em-ployed to implement this identification procedure. Ifthe bulk contribution η0 is not identified, the Preisachweight function µ has to contain this constant contri-bution, which can cause a significant degradation ofperformance of the identified model.

The discrete Preisach model is defined by two pa-rameters, the discretization level nh and the range(βm, αm). During an identification process it is impor-tant that the range of the input signal and the rangeof the Preisach model correspond. In the rest of thispaper it is assumed that all Preisach elements have aninitial state of −1, i.e. in the S− region, and that theinput resembles the increasing signal shown in Fig. 6a.This ensures that the Preisach elements correspondingto small (β, α) values are excited first, due to the ini-tial negative input. Below, three different input rangesare employed in identification of a Preisach model withdiscretization level nh = 15, βm = −90 and αm = 90.

• In Fig. 7a a too small input range is applied for thePreisach model. As a result of this, the end diag-onal elements in the Preisach distribution is zero.Moreover, such an input range does not excite anyborder element of the distribution more than max-imally once (depending on if all Preisach weightsare defined as positive or negative initially), mak-ing the model store some of the static bulk contri-bution η in these border elements.

• By applying a more suitable input range, i.e. [-91,91], all elements of the Preisach distribution areexcited, illustrated in Fig. 7b. This also removes

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Input Voltage [V]

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Figure 8: Model output when identifying using a 20 Hz sine wave, with 90 V amplitude, with nh = 20 asdiscretization level.

the stored static contribution seen in the border el-ements in the previous case. To excite all elementsin the discrete Preisach model, the maximum am-plitude of the input signal has to exceed αm forthe states to switch from -1 to 1. This is achievedwith the input range in this case.

• When employing a too large input range, as shownin Fig. 7c, the end diagonal elements have to con-tain the weights for all inputs exceeding (βm, αm).This will make the model a poor fit for all inputvalues larger than the Preisach model range.

An identified Preisach model will, when providedwith an input signal, produce a hysteretic output. Thisshould be done to validate that the identified modelcaptures the measured hysteresis behavior. In Fig. 8athis verification has been done on a sinusoidal signal of20 Hz, using a Preisach model with discretization levelnh = 20. The hysteresis loop between the input volt-age and output measurement can be seen in Fig. 8b.These plots also illustrate the discrete nature of thePreisach model, where the output changes wheneverthe values of the delayed relay elements switches. Thesteps of the model output vary in step size, resultingfrom different Preisach weights in the identified model.

The discrete behavior of the Preisach operator, dis-cussed above, creates an error between the measuredand modelled output. To reduce this error, the dis-cretization level of the model can be increased, whichwill improve the fit between the measured output andthe model output. Concerns related to a higher dis-cretization level is that the model order increases,which in turn makes both identification and inversionmore time consuming.

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Inp

ut

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tage

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Figure 9: Illustration of the proposed linear interpola-tion applied in the inverse method.

4 INVERSION

4.1 Inversion Algorithm and Interpolation

As previously mentioned, to compensate for hystere-sis the Preisach model must be inverted. However thedelayed relay operator does not have an analytical in-verse, and therefore requires a numerical approxima-tion. In this paper a closest match algorithm is im-plemented, exploiting the discrete nature of the modelby finding the input value which produces an input asclose as possible to the reference signal. Due to theclosest match property, the inverted signal will alwayshave the optimal value based on the discrete model.The implementation of the closest match algorithm isoutlined in more detail in Tan et al. (2001).

The inverted input of the discrete Preisach modelnaturally has a discrete behavior, which introduces

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high frequencies in the inverted signal in the discretesteps. These frequencies can excite the vibration dy-namics in stiff structures with little damping, such asdevices using piezoelectric actuators, causing signifi-cant loss of precision in hysteresis compensation. Con-sequently, to avoid these vibrations, the input shouldbe smoothed, which can be performed by, for instance,linear interpolation as proposed in Tan et al. (2001).In this paper, a specific method for choosing interpo-lation points is proposed, illustrated in Fig. 9, whereeach point to interpolate is chosen at a discrete stepof the inverted signal. The discrete input in the figurerefers to the inverted signal without interpolation. Theresulting inverted input signal significantly reduces themagnitude of the higher frequency components (result-ing from the discrete steps), subsequently attenuatingthe excitation of the vibration dynamics. The perfor-mance of a few other interpolation methods are studiedin Stakvik (2014).

By employing interpolation for smoothing, the hys-teresis compensation scheme cannot be applied for real-time implementation with closed loop control. The in-terpolation procedure requires the next step of the dis-crete input, however since the reference signal is calcu-lated in real-time of a controller the interpolation willbe affected by a varying time-delay. This time-delay isdue to the uncertainty of the future reference signalsfor the inversion algorithm. This issue can be circum-vented by applying a direct feed-forward compensationfrom the reference trajectory, and in this way avoidingthe varying time-delay.

4.2 Experimental Results of HysteresisCompensation

If a suitable input range is applied in the identifica-tion of the model, experimental tests of hysteresis com-pensation can be performed. This section will exem-plify some practical issues concerning the choice of dis-cretization level and reference frequency for hysteresiscompensation. The following three scenarios will illus-trate the consequence of a too small discretization level,a too high reference frequency, while the last scenarioexemplifies the best performance of hysteresis compen-sation achieved by applying the Preisach model. Forall experiments an input with 100 reversals is used foridentifying the models. The reference signal is a sinu-soidal with 10 µm amplitude, with the frequencies ofthe input signal defined for each scenario.

4.2.1 Low Discretization

Applying a low discretization level on the Preisachmodel is desirable due to low computational effort inidentification and inversion. Fig. 10 shows hysteresis

compensation based on a model with a discretizationlevel nh = 10 and reference input as a sinusoidal signalof 5 Hz. The error between the reference and the mea-surement is significant, in addition, the measurementlags behind the reference, i.e. it does not compensatefor the hysteresis sufficiently. By comparing the mea-surement in Fig. 10 with the interpolation method inFig. 9, the same decreasing ramp is observed in themaximum of the reference. This implies that for lownh the interpolation method causes a significant errorin the extremums of the reference.

Additionally, the low discretization level introducesan error in the extremum of the reference that cannot be explained by the interpolation method. Thiserror originates from a poor model description of thehysteresis, where the output is modelled incorrectly forthese values, despite the fact that the identification wasconducted with a larger input range than the maximumamplitude of the reference signal. If the discretizationlevel is increased the model description should be ableto capture these variations more precisely.

4.2.2 High Frequency

Due to the poor performance of the low discretizationlevel above, the discretization is increased to nh = 100.The interpolation procedure was introduced to avoidlarge frequency components in the input signal due tosteps in the original signal. However if the frequencyof the reference signal is increased, the frequency com-ponents of the system also increase, causing vibrationsin the piezoelectric actuator. These vibrations are il-lustrated in Fig. 11 where the reference signal has afrequency of 50 Hz. The measurement reveals that thevibrations have a frequency of about 750 Hz, corre-sponding to the resonant dynamics of the piezoelectricactuator. Even though there are significant vibrationsin the measurement, the amplitude of the error is con-siderably smaller than for the previous case. This ismainly due to the increase of the discretization level.

In general, vibrations as in Fig. 11, are present whenthe reference frequency is lower. However for a refer-ence signal with sufficiently low fundamental frequency,the magnitude of the frequency components, of the in-verted input around the resonance frequency, is suffi-ciently low for the vibrations to be dominated by othersources of noise. To apply reference signals with highfundamental frequencies, some kind of vibrational com-pensation scheme must be performed.

4.2.3 High Discretization and Low Frequency

By combining the high discretization level of nh = 100with low reference frequency of 5 Hz, a best perfor-mance of hysteresis compensation using the Preisach

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Figure 12: Best performance of Preisach model hysteresis compensation with discretization level nh = 100.

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Figure 11: Hysteresis compensation with a too highreference frequency for Preisach model withdiscretization level nh = 100, where the vi-brations are caused by actuator dynamics.

model is illustrated in Fig. 12. From Fig. 12bthe reference-measurement relationship shows that thehysteresis is compensated, and from Fig. 12a, the mea-surement follows the reference without lagging behind.The error between reference and measurement is ap-proximately 5% at the maximum amplitude, whichcan be due to varying environmental conditions, e.g.temperature variations of the piezoelectric material,between the time of identification and compensation.Another reason for the error can be due to the factthat there are some inaccuracies in the identified modelclose to the extremums of the reference. This hypoth-esis is supported by the fact that the measurement hasa larger amplitude than the reference. For hysteresiscompensation using the Coleman-Hodgdon model, theerror has been shown to be less than 1% Eielsen et al.(2012). When comparing with these results, compen-sation of hysteresis with the Preisach model performspoorly, however, an adaptive identification procedurecould increase the compensation performance.

5 CONCLUDING REMARKS

This paper presents implementational issues of hystere-sis compensation by employing the Preisach model,based on the delayed relay operator. Suggestions ofhow to avoid these issues are presented to obtain a sat-isfactory performance of the hysteresis compensation.

In particular, aspects of the identification and inver-sion procedure are shown, and the effect of differentmodel parameters and input ranges are illustrated. Inthe discrete model, the maximum input range shouldexceed the range of the Preisach model with a smallamount. A proposal of how to estimate both thePreisach weight function and the bulk modulus wasgiven, and results from an identification were shown.

Further, the inversion procedure was explained, fo-

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cusing firstly on the need for interpolation of the in-verted signal to avoid unnecessary high frequencies. Aspecific method for choosing interpolation points wasproposed and illustrated. Issues with low discretizationlevels and high frequency reference signals were illus-trated with experimental results, showing significanterrors and vibrations, respectively. At last, a best per-formance of hysteresis compensation by applying thePreisach model was presented.

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