System Identification Using Associated Linear Equations

Mechanical Systemsand

Signal Processing

www.elsevier.com/locate/jnlabr/ymssp

Mechanical Systems and Signal Processing 18 (2004) 431–455

System identification using associated linear equations

J.A. Vazquez Feijoo*, K. Worden, R. Stanway

Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street,

Sheffield S1 3 JD, UK

Received 24 July 2002; received in revised form 14 May 2003; accepted 19 May 2003

Abstract

In this paper, analytical models of non-linear systems are obtained by identifying the frequency responsefunctions (FRFs) of their associated linear equations (ALEs). This allows the use of several methods ofidentification in the frequency domain usually applicable to linear systems. Among other advantages, thecumbersome multidimensional Fourier Transformation required in higher-order frequency responsefunctions (HFRFs) analysis is eliminated. Two theoretical systems are used here as examples, anelectrostrictive actuator and a Duffing oscillator. The concept of the non-linear gain constant arises as asimple means of identification.r 2003 Elsevier Ltd. All rights reserved.

Keywords: Non-linear systems; System identification; Volterra series; Higher-order frequency response function;

Associated linear equations

1. Introduction

The higher-order frequency response functions (HFRFs) describe in the frequency domain, thecharacteristics of time-invariant continuous non-linear systems. Although a very convenientanalytical tool, their practical application is limited to low levels of excitation because higher-order functions require multidimensional Fourier transformation. Obtaining HFRFs of ordershigher than the third is time-consuming and the results are often difficult to interpret.

In previous work, these systems were modelled by a sequence of continuous linear modelsnamed associated linear equations (ALEs) [1]. When the appropriate input signal is used, theoutput of each ALE is the corresponding order Volterra operator. The relationship between theVolterra operators and the HFRFs are well established. Because all the ALEs are linear models,

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*Corresponding author.

E-mail address: [email protected] (J.A. Vazquez Feijoo).

0888-3270/03/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0888-3270(03)00078-5

they each have a standard linear response function that is named here the associated frequencyresponse function (AFRF) to emphasise that they correspond to a specific order ALE. To avoidthe technical difficulties in obtaining the HFRFs, it is proposed here to find the AFRF of eachALE. From the AFRF of the nth-order ALE, the corresponding HFRF of the original system isobtained straightforwardly. Also, the associated AFRF is easier to interpret and provides thesame information as the HFRF [1]. For any order, by the appropriate spectra from the system, itis possible to identify the AFRF and then the nth-order ALE. Thereafter, when all the Volterraoperators are added, an analytical model is obtained.

In this paper, two single-input single-output (SISO) theoretical systems are used as examples, aHammerstein cascade which models an electrostrictive actuator and a Duffing oscillator. Once thespectra are obtained, the system parameters are extracted by a simple curve-fitting method. Fromthe simulation results on both examples, it is emphasised that the difference between the first-order ALE and all other superior orders is just the amplitude of the output signal. Therefore, thenon-linear problem identification is reduced to finding a single scalar named here the non-lineargain constant Kn; for each order. This parameter can be obtained by only a one-dimensionalFourier transformation independent of the ALE order.

2. Theory

For a given non-linear system, using Gaussian white noise as the input and evaluating theappropriate spectra, an estimation of the first two higher-order frequency response functions(HFRFs) can be obtained [2],

H1ðoÞDL1ðoÞ ¼SyxðoÞSxxðoÞ

; ð1Þ

H2ðo1;o2ÞDL2ðo1;o2Þ ¼Sy0xxðo1;o2Þ

2!Sxxðo1ÞSxxðo2Þ; ð2Þ

where Sxy is the cross power spectrum between the variables yðtÞ and xðtÞ; where xðtÞ representsthe system input and yðtÞ represents the system output. The prime on y indicates that the mean ofthe signal has been removed. In theory, any HFRF can be obtained by the same procedure. Inpractice, the limitation is on the order of the multidimensional Fourier transformation that can beundertaken.

In [1] it was established that if a non-linear system possesses a convergent Volterrarepresentation,1 it can be split into a series of linear subsystems acting in parallel (each oneproducing a specific order Volterra operator). Those systems are modelled by ALEs. The non-linear response, which for continuous non-linearities can be represented by polynomial terms, isrestricted in the ALE models to act as the excitation in the linear subsystems.

Because the subsystems are linear, by the appropriate selection of the input (the appropriatenon-linear terms), Eq. (1) can be applied to any higher frequency order ALE and then by only aunidimensional Fourier transformation obtain a linear spectrum named here as the AFRF. For

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1Here for convergence it is understood that a finite number of operators exactly represents the system or at least a

truncated series mimics the system up to a predefined accuracy.

J.A. Vazquez Feijoo et al. / Mechanical Systems and Signal Processing 18 (2004) 431–455432

an SISO linear structural system, the parameters to evaluate can be considered to be reduced tothree, the natural frequency o; the damping ratio z or the lost factor (in case of hystereticdamping). The methods presented here can be carried out in two steps, the parameter estimationof the linear part and a second step that is the identification of the non-linear terms that have to beincluded in the model. The trial input is a product of the non-linear terms to be tested; if aspectrum signal is obtained then it is assumed that this term is part of the system excitation of acertain frequency order.

The non-linearity can appear either in the input (i.e. Hammerstein models) or in the output (theDuffing oscillator is a classic example). An example of each case is considered here. In bothmodels viscous damping is assumed, therefore the Kelvin–Voigt [3] model describes the generalstructure linear system and therefore becomes the skeleton of the ALEs

.y1 þ 2zon ’y1 þ o2ny1 ¼ AuðtÞ; ð3Þ

where uiðtÞ is the trial input in the ALE (for the first-order ALE, u1ðtÞ is the real system excitation).The equivalent expression for this model in the frequency domain is

H1ðoÞ ¼A

�o2 þ i2zonoþ o2n

: ð4Þ

With respect to the Hammerstein model, its general polynomial form is

.y þ 2zon ’y þ o2ny ¼

Xn

i¼1

aixi: ð5Þ

In [4], the ALEs for the structural Hammerstein model have to be

.yi þ 2zon ’yi þ o2nyi ¼ aix

i: ð6Þ

When identifying a system, the trial terms for testing if there is a non-linearity in the input of ithorder is uiðtÞ ¼ xi: In particular, the second-order ALE for the case that there is second-order non-linearity coming only from the input is2

.y2 þ 2zon ’y2 þ o2ny2 ¼ a2x2: ð7Þ

As the relationship between u2ðtÞ and y2ðtÞ is linear, according to Eq. (4), the FRF is

H12ðO2Þ ¼a2

�O22 þ i2zonO2 þ o2

n

; ð8Þ

where O2 ¼ o1 þ o2; as it is valid for linear systems, Eq. (1) can be used to obtain H12ðO2Þ:

H12ðO2ÞEL12ðO2Þ ¼Sy2u2

ðO2ÞSx2u2

ðO2Þ¼

Sy2x2ðO2ÞSx2x2ðO2Þ

ð9Þ

the modal analysis extract from this signal the parameters value.3 The second-order HFRFfor the Hammerstein model can be obtained by Harmonic probing as in [5] when applied to

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2Note that x/yi is not a linear system. However, ui/yi is.3 It is known [1] that the natural frequency on and the damping ratio z for each ALE are of the same value. However,

in the rest of this work it is supposed that each order possesses its own values of these parameters. This is with the

intention of obtaining several estimations of these two parameters.

J.A. Vazquez Feijoo et al. / Mechanical Systems and Signal Processing 18 (2004) 431–455 433

Eq. (7) yielding

H2ðo1;o2Þ ¼a2

�ðo1 þ o2Þ2 þ i2zonðo1 þ o2Þ þ o2

n2

: ð10Þ

Comparison with Eq. (9) gives H2ðo1;o2Þ ¼ H12ðO2Þ and only a first-order Fourier transform isneeded to obtain the second-order HFRF or in fact any other order observe that in general for theHammerstein cascade,

Hnðo1;o2;y;onÞ ¼ H1nðOnÞESynxnðOnÞSxnxnðOnÞ

ð11Þ

with On ¼ o1 þ o2 þ?þ on: When the non-linearity belongs to the output something similar ispossible. Consider the Duffing oscillator,

.y þ 2zon ’y þ o2ny þ

k2

my2 þ

k3

my3 ¼

1

mx: ð12Þ

From [1], the second-order ALE is found to be

.y2 þ 2zon ’y2 þ o2ny2 ¼ �

k1

my21: ð13Þ

If u2 ¼ y21; the third-order AFRF can be obtained from Eq. (1) as

H12ðO2ÞESy3y2

1ðO2Þ

Sy31y21ðO2Þ

: ð14Þ

Harmonic probing applied to Eq. (13) gives

H12ðO2Þ ¼�k2=m

�O22 þ i2zonO2 þ o2

n

ð15Þ

and to Eq. (12) gives

H2ðo1;o2Þ ¼�ðk2=mÞH1ðo1ÞH1ðo2Þ

�ðo1 þ o2Þ2 þ i2zonðo1 þ o2Þ þ o2

n

: ð16Þ

From harmonic probing on the first ALE one has

H1ðoÞ ¼�1=m

�o2 þ i2zonoþ o2n

: ð17Þ

If O2 ¼ o1 þ o2; then,

H2ðo1;o2Þ ¼ H12ðO2ÞH1ðo1ÞH1ðo2Þ: ð18Þ

From Eq. (12), it is clear that the non-linear behaviour comes from the non-linear termscontaining the k2 and k3; respectively. It is then expected that the excitation of any higher-orderALE have to be any term of the polynomial expansion of yðtÞ when expressed by its Volterraoperators, i.e.,

k2

myðtÞ2 ¼

k2

mð y1ðtÞ

2 þ 2y1ðtÞy2ðtÞ þ 2y1ðtÞy3ðtÞ þ?þ y2ðtÞ2

þ 2y2ðtÞy3ðtÞ þ?Þ ð19Þ

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and

k3

myðtÞ3 ¼

k3

mð y1ðtÞ

3 þ 3y1ðtÞ2y2ðtÞ þ 3y1ðtÞy2ðtÞ

2 þ?þ y2ðtÞ3

þ 3y2ðtÞ2y3ðtÞ þ?Þ: ð20Þ

Many methods have been developed for extracting the system parameters from the spectralanalysis of linear structural systems. Any of them, when appropriate for the particular linearspectrum, can be used to identify each AFRF. As the examples provided here are SISO, a simplecircle-fitting method is used [6].

In general, for a linear Kelvin–Voigt SISO structural system the mobility is

a1ðoÞ ¼iAo

�o2 þ 2zonoi þ o2n

: ð21Þ

The mobility Nyquist plot (Fig. 1) is the familiar circle from which the system parameters can beextracted. The natural frequency on is the frequency diametrically opposite the origin. Thedamping ratio can be obtained from

z ¼ðoa � obÞ

2on

; ð22Þ

where oa and ob are any two frequencies that subtend the same angle y from the naturalfrequency on to the left and to the right (Fig. 1). Finally, from the diameter D of the mobilitycircle, the modal constant is obtained as

A ¼ 2zonD: ð23Þ

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Fig. 1. Mobility circle from the Nyquist plot.


3. Identification of a Duffing oscillator by ALEs

3.1. Description of the Duffing oscillator

The Duffing oscillator is modelled as a single-degree-of-freedom (SDOF) soft spring–mass–damper system; the non-linear spring force is obtained as

Fk ¼ �2:5� 107dþ 25� 1010d3: ð24Þ

The mass is 10 kg and the viscous damping coefficient is 1000 N=ðm=sÞ: The equation of motion isidentical to Eq. (12) with k2 ¼ 0: When the data are substituted in this equation,

10 .yðtÞ þ 1000 ’yðtÞ þ 2:5� 107yðtÞ � 2:5� 1010y3 ¼ xðtÞ: ð25Þ

From Eq. (25) it is possible to extract directly the natural frequency and the damping ratio:

o2n ¼ 2:5� 107 N=kg m-on ¼ 1581:1 rad=s; ð26Þ

2zon ¼ 100 N s=kg m-z ¼ 0:0316: ð27Þ

As there is no even power of yðtÞ in Eq. (25), all even Volterra operators vanish. And only theterms included in Eq. (20) are going to be the excitation to the higher-order ALEs. From Eq. (12),the lower non-zero order ALEs can be obtained as in [1]:

.y1ðtÞ þ 100 ’y1ðtÞ þ 2:5� 106y1ðtÞ ¼ 0:1xðtÞ; ð28Þ

.y3ðtÞ þ 100 ’y3ðtÞ þ 2:5� 106y3ðtÞ ¼ 25� 109 y31ðtÞ; ð29Þ

.y5ðtÞ þ 100 ’y5ðtÞ þ 2:5� 106y5ðtÞ ¼ 7:5� 109y21y3ðtÞ; ð30Þ

.y7ðtÞ þ 100 ’y7ðtÞ þ 2:5� 106y7ðtÞ ¼ 7:5� 109ð y21y5ðtÞ þ y2

3y1ðtÞÞ: ð31Þ

In analogy with the first-order Kelvin–Voigt model (Eq. (3)) in which A is the input amplitude, theinput amplitude of nth-order ALE ai is according to the following equations:

a1 ¼ 0:1 kg�1;

a3 ¼ 2:5� 109 rad=s2 m2;

a5 ¼ 7:5� 109 rad=s2 m2;

a7 ¼ 7:5� 109 rad=s2 m2:

3.2. Identification process

The ALEs are going to be found in ascendant frequency order. The first-order response isobtained as follows; the excitation level is adjusted until a linear response is obtained (see Section 6).For the present example a band-limited white noise is used (though a random signal can providethe appropriate spectrum). Eq. (1) gives the first-order AFRF in receptance form. In order toobtain the mobility the receptance is multiplied by io: Fig. 2 shows the circle obtained. Thespectra is obtained from 100 averages of groups of 1024 points; a Hanning window is used.

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The parameters obtained from circle fitting procedure are

a1 ¼ 0:0956 kg�1; z ¼ 0:0307 and on ¼ 1565 rad=s:

According to Eq. (4), the first-order AFRF in receptance form is4 then

H1ðoÞ ¼0:0956

�o2 þ 96:14oi þ 2:49331� 106: ð32Þ

Applying the inverse Fourier transform gives the first-order ALE:

.y1ðtÞ þ 96:1914 ’y1ðtÞ þ 2:4933� 105y1ðtÞ ¼ 0:0956xðtÞ: ð33Þ

The next step is to identify the second order. At this point of the identification process, thepolynomial non-linear terms that have to be added to the model are unknown. Therefore, it isnecessary to test all the possible terms that can produce a second-order response. If the non-linearity comes from the input, the unique possible non-linear term is

u2 ¼ x2 ð34Þ

and if the non-linearity comes from the output,

u2 ¼ y21: ð35Þ

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Fig. 2. Circle fitted to the mobility Nyquist plot of the first-order Duffing oscillator response.

4Note that the corresponding H11ðoÞ is actually the well-known linear FRF; so for this first order there is no need to

repeat sub-indexes, and it should be no source of confusion. Denote this function as H1ðoÞ:


Trying the input from Eq. (35) Eq. (1) is for this case

H12ðO2ÞEL1ðO2Þ ¼Sy2ð y2

1ÞðO2Þ

Sð y21Þð y2

1ÞðO2Þ

: ð36Þ

The results obtained for the second-order AFRF are shown in Fig. 3. The real and imaginaryparts are observed to have zero mean. This behaviour suggests correctly that there is no second-order excitation from the output ð y2ðy2

1ðtÞÞ ¼ 0Þ: The mobility Nyquist plot corresponding toFig. 4 shows that there is no circular structure. A similar result is obtained when the input fromEq. (34) is tested. Therefore there is no second-order response from the system (as this is what isexpected when observing Eq. (25)).

Considering the third-order response from Eqs. (19) and (20), the input into the third-orderALE is

u3ðtÞ ¼ x3ðtÞ ð37Þ

or

u3ðtÞ ¼ y1ðtÞ3 þ 2y1ðtÞy2ðtÞ: ð38Þ

As long as there is no second-order component, only the first term contributes to the third-orderresponse of the system; this leads to

H13ðO3ÞEL1ðO3Þ ¼Syð y3

1ÞðO3Þ

Sð y31Þð y3

1ÞðO3Þ

: ð39Þ

Fig. 5 shows a clear linear third-order response and the mobility Nyquist plot (Fig. 6) exhibits atrue circular shape. Using the same procedure as in the first-order AFRF, the parametersextracted for the third-order ALE are

a3 ¼ 2:659� 109 rad=s2 m2; z ¼ 0:0ð26Þ8; on ¼ 15068:2 rad=s:

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Fig. 3. From Eq. (43), spectra obtained for the second-order response of the Duffing oscillator.


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Fig. 4. Second-order mobility Nyquist plot of the Duffing oscillator.

Fig. 5. AFRF of the third-order ALE.


The corresponding third-order AFRF is

H13ðO3Þ ¼2:659� 109

�O23 þ 110iO3 þ 2:5033� 106

ð40Þ

and the ALE

.y3ðtÞ þ 110 ’y3ðtÞ þ 2:5033� 106y3ðtÞ ¼ 2:659� 109y1ðtÞ3: ð41Þ

Harmonic probing on Eq. (25) gives the third-order HFRF as

H3ðo1;o2;o3Þ ¼2:659� 109H1ðo1ÞH1ðo2ÞH1ðo3Þ

�ðo1 þ o2 þ o3Þ2 þ 110iðo1 þ o2 þ o3Þ þ 2:5033� 106

: ð42Þ

The next ALE for which a true response has been found is of fifth order. The unique responsecomes from (Eq. (20)) as

u5ðtÞ ¼ y1ðtÞ2y3ðtÞ: ð43Þ

The AFRF of the fifth ALE is

H15ðO5Þ ¼Sy5ð y2

1y3ÞðO5Þ

Sð y21y3Þð y2

1y3ÞðO5Þ

: ð44Þ

The corresponding extracted parameters are

a5 ¼ 7:4725� 109 rad=s2 m2; z ¼ 0:0363; on ¼ 1558:3 rad=s:

So the fifth-order ALE and its AFRF are

H15ðO5Þ ¼7:4865� 109

�O25 þ 114O5i þ 2:47121� 106

; ð45Þ

.y5ðtÞ þ 114 ’y5ðtÞ þ 2:47121� 106y5ðtÞ ¼ 7:4865� 109y1ðtÞ2y3ðtÞ: ð46Þ

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Fig. 6. Mobility circle of the AFRF of the third-order ALE of the Duffing oscillator.


From Eq. (25), using harmonic probing, the fifth-order HFRF is

H5ðo1;o2;o3;o4;o5Þ ¼ H15ðo1 þ o2 þ o3 þ o4 þ o5ÞH1ðo1ÞH1ðo2ÞH3ðo3;o4;o5Þ: ð47Þ

Now, in order to measure how well the truncated Volterra series represents the system the meansquare error (MSE) is used [8]

MSE ¼100

Ns2n

XN

i¼1

ðni � #niÞ2: ð48Þ

Here N is the number of data points, s2n the variance of n that is the ‘‘true’’ value and #n is the

‘‘approximate value’’ that for the present case is provided by the truncated Volterra series. TheMSE considers the variances and error of the data. The fifth-order Volterra representation givesan error of just 0.2953% for an excitation level up to 0:35 MN:

4. Identification of an electrostrictive actuator by ALEs

4.1. The electrostrictive-system parameters

Following Appendix A, the electrostrictive actuator is represented by the equation

.y þ 9128:7 ’y þ 2:0833� 103y ¼ 2:4306� 10�4x2 � 7:027� 10�22x6: ð49Þ

As in the square law version this system exhibits no first-order component in the output. Toobtain a first-order Volterra operator, a DC component is going to be added at the output. Thiscomponent is obtained by adding a constant signal a in the input [4]; the actual signal uðtÞ into theelectrostrictive system is

uðtÞ ¼ xðtÞ þ a; ð50Þ

where a is an appropriate constant that for this case is a ¼ 8 kV:

.y þ 9128:7 ’y þ 2:0833� 103y ¼ 2:4306� 10�4ðx þ aÞ2 � 7:027� 10�22ðx þ aÞ6: ð51Þ

Developing both parentheses in Eq. (51), one obtains

.y þ 9128:7 ’y þ 2:0833� 103y

¼ �7:027� 10�22x6 þ 6ð2:4306� 10�4Þax5 þ 15½�7:027� 10�22a2x4

þ 20ð�7:027� 10�22Þx3a3 þ ð2:4306� 10�4 þ 15½�7:027� 10�22a4Þx2

þ ½2ð2:4306� 10�4Þaþ 6ð�7:027� 10�22Þa5x

þ ð2:4306� 10�4a2 þ ð�7:027� 10�22Þa6Þ: ð52Þ

This system possesses a finite Volterra representation of sixth order [4]. As just up to the fourthorder is going to be identified here, the corresponding ALEs are

y0 ¼ 2:11� 10�2; ð53Þ

.y1 þ 9128:7 ’y1 þ 2:0833� 103y1 ¼ 3:75x; ð54Þ

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.y2 þ 9128:7 ’y1 þ 2:0833� 103y1 ¼ 1:9989� 10�4x2; ð55Þ

.y3 þ 9128:7 ’y3 þ 2:0833� 103y3 ¼ 1:9989� 10�43 x3; ð56Þ

.y4 þ 9128:7 ’y4 þ 2:0833� 103y4 ¼ �6:7459� 10�13x4: ð57Þ

From Eqs. (53)–(57), the amplitudes to identify are

a0 ¼ 2:11� 10�2 m=s2; a1 ¼ 3:75 m=s2 V;

a2 ¼ 1:9989� 10�4 m=s2 V2; a3 ¼ 1:9989� 10�4 m=s2 V3;

a4 ¼ �6:7459� 10�13 m=s2 V4:

4.2. Identifying the electrostrictive system

For this system, once more the input to the system is a band-limited zero-mean white noise. Asin the case of the Duffing oscillator, an amplitude at which the system responds linearly is selected(Section 6). From here the next step is to apply Eq. (1) to the data. Observe that although theinput is zero-mean, the output possesses a DC component. This mean component is the responseto the a which is part of the system. This response is taken into account by a constant y0 that is thezero-order Volterra operator.

To evaluate y0; the system output is measured when there is no excitation into the systemðxðtÞ ¼ 0Þ obtaining

y0 ¼ 7:3632� 10�6 m=s2: ð58Þ

The next step is to excite the system by the appropriate amount, to estimate the first-orderoperator from the output data (Refer to Section 6). Once the linearity is assured, Eq. (1) isapplied. The parameters extracted from the Nyquist plot (Fig. 7) are

a1 ¼ 3:3944 m=s2 V; z ¼ 0:1171; on ¼ 44638 rad=s:

The electrostrictive actuator first-order ALE and its corresponding AFRF when estimatedaccording to Eq. (3) are

.y1ðtÞ þ 1:0579� 104 ’y1ðtÞ þ 1:9926� 109y1ðtÞ ¼ 3:3944xðtÞ; ð59Þ

H1ðoÞ ¼3:3944

�o2 þ 10579ioþ 1:9926� 109: ð60Þ

The second-order AFRF is once more tested for u2ðtÞ ¼ x2ðtÞ and u2ðtÞ ¼ y21ðtÞ: It is found that

only the former produces a second-order response as shown in Fig. 8. The parameters extractedare

a2 ¼ 1:6394� 10�9 m=s2 V2; z ¼ 0:1278; on ¼ 43769 rad=s:

The second-order ALE and AFRF according to Eq. (3) are

.y2 þ 11346:4 ’y2 þ 1:928� 109y2 ¼ 1:7794� 10�4x2; ð61Þ

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H12ðO2Þ ¼1:7794� 10�4

�O22 þ 11346iO2 þ 1:928� 109

: ð62Þ

According to Eq. (7), the corresponding HFRF is

H2ðo1;o2Þ ¼ H12ðO2Þ ¼1:7794� 10�9

�ðo1 þ o2Þ2 þ 11346iðo1 þ o2Þ þ 1:928� 109

: ð63Þ

For the third-order ALE, the unique term found that produces a response of this order is u2ðtÞ ¼xðtÞ3; and the results are

a3 ¼ 7:23� 10�9 m=s2 V3; z ¼ 0:11; on ¼ 43725 rad=s;

so,

.y3 þ 9653:4 ’y3 þ 1:9254� 109y3 ¼ �7:23� 10�9x3 ð64Þ

and the third-order HFRF is

H3ðo1;o2;o3Þ ¼ H13ðO3Þ ¼�7:23� 10�9

�ðo1 þ o2 þ o3Þ2 þ 9653iðo1 þ o2 þ o3Þ þ 1:9254� 109

: ð65Þ

The fourth-order parameters, ALE and its corresponding HFRF are

A4 ¼ 6:6768� 10�13 m=s2 V3; z ¼ 0:1168; on ¼ 44533 rad=s;

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Fig. 7. Circle fitting of the first-order harmonic response of the electrostrictive actuator.


.y4 þ 10528 ’y4 þ 1:9832� 109y4 ¼ �6:6908� 10�13x4; ð66Þ

H4ðo1;o2;o3;o4Þ

¼ H14ðO4Þ ¼�6:6908� 10�13

�ðo1 þ o2 þ o3 þ o4Þ2 þ 10528iðo1 þ o2 þ o3 þ o4Þ þ 1:9832� 109

: ð67Þ

The electrostrictive actuator from which the data were modelled suffers saturation at 15 kV: Thetheoretical model presented here does not consider saturation. This is in order to be able to assessup to which input amplitude a certain order Volterra system provides an accurate representation.Here, the fourth-order Volterra representation proved to be accurate up to a 1% mean squareerror for an input amplitude of 20 kV: This voltage is higher than the saturation, therefore nohigher-order Volterra representation may be needed.

5. The non-linear gain constant

From the results of the two cases presented here, and considering that variations in theparameter estimates appear as a consequence of the curve-fitting process, all the ALEs show thesame natural frequency on and the same damping ratio z as the theory predicts [1]. Table 1

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Fig. 8. Circle fitting of the second-order harmonic response of the electrostrictive actuator.


contains the average value of the ALE parameters for both cases and the mean square errorcompared to the actual parameters.

As on and z are the same, independent of the ALE order (e.a. observe Eqs. (16) and (17)),

H1nðoÞH1ðoÞ

¼an

a1¼ Kn; ð68Þ

where Kn defined here, is the non-linear gain constant, which can be used for identification asfollows. For a given system, two different inputs xðtÞ and x0ðtÞ are tried. From the first excitationxðtÞ; the nth-order AFRF is according to Eq. (1):

H1nðOnÞ ¼Synun

ðOnÞSunun

ðOnÞ; ð69Þ

where un is the excitation to the nth-order ALE produced when the system input is xðtÞ: At thisstage of the identification process, the first-order ALE is already known, then this first-orderresponse can be simulated as an independent linear system. The second excitation x0ðtÞ goes intothis system and is made numerically equal to x0ðtÞ ¼ un: The corresponding first-order AFRF isobtained as

H 01ðOnÞ ¼

Sy01unðOnÞ

SununðOnÞ

: ð70Þ

As the system parameters do not depend on the input and just on the system characteristics, whenEqs. (69) and (70) are substituted into Eq. (68), the non-linear gain constant Kn can be obtained as

Kn ¼Synun

ðOnÞ=SununðOnÞ

Sy01unðOnÞ=Sunun

ðOnÞ; ð71Þ

so,

Kn ¼Synun

ðOnÞSy0

1unðOnÞ

: ð72Þ

Once more all the possible terms of second order, third, fourth etc., have to be tested. From hereon only the terms that produce an nth-order output component are going to be mentioned. Forthe second order

K2 ¼Sy2x2ðOnÞSy0

1x2ðOnÞ

: ð73Þ

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Table 1

Average of the system parameters obtained from the ALEs

Duffing oscillator Electrostrictive actuator

Average e% True value Average e% True value

on 1577.8 0.2 1581.1 44239.75 3.1 45643.54

z 0.03293 4.2 0.0316 0.119 19 0.1


The first step is to estimate y1ðtÞ: The second step is to make x0ðtÞ ¼ u2 ¼ x2ðtÞ and analyticallycompute y01½u2ðtÞ: Then apply Eq. (72). The inputs for the third and fourth order are, respectively,

x0ðtÞ ¼ xðtÞ3 ð74Þ

and

x0ðtÞ ¼ xðtÞ4: ð75Þ

The results from the simulation are plotted in Figs. 9 and 10. The non-linear gain constant Kn isobtained from a mean value calculated around the resonance zone. The same figures include twoextra lines that show the 95% confidence that the true value falls into the interval enclosed bythese lines. This interval is calculated as if the spectral distribution were Gaussian. If kn is thespectra mean, then,

�1:96spKn � knp1:96s;

where s is the standard deviation of the spectrum.According to Eq. (68), the amplitude ratio an can then be obtained as

an ¼ Kna1: ð76Þ

Table 2 shows the Kn obtained and its corresponding an: Observe that the true value resides withinthe 95% confidence.

The non-linear constants of the Duffing oscillator are obtained by the same procedure.

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Fig. 9. Results obtained from Eq. (76) for the second-order non-linear gain constant K2 of the electrostrictive actuator.


Figs. 11 and 12 show a sample of the results for the Duffing oscillator and Table 3 shows theamplitude ratio an obtained from Kn:

6. Obtaining the appropriate amplitude for the Nth-order spectral response

This section deals with the criteria to select the proper amplitude for isolating the higher-orderresponses. In both methods used here, the process of identification goes from the lower to thehigher-order response. In order to obtain the first Volterra operator, an amplitude should befound that guarantees that the system response is dominantly linear. There are severalmethodologies to determine if the system is linear or not, i.e. the correlation function [7,9] orthe Hilbert transformation [6]. The former has some restrictions that prevent it from being used

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Fig. 10. Results obtained from Eq. (76) for the third-order non-linear gain constant K3 of the electrostrictive actuator.

Table 2

Non-linear gain constants of the electrostrictive actuator and the amplitude ratio obtained from them

Second-order Third-order Fourth-order

Kn calculated 5:2075� 10�5 �2:1779� 10�9 2:0708� 10�13

Kn real 5:3304� 10�5 �1:91� 10�9 �1:799� 10�13

an calculated 1:7915� 10�4 �8:6� 10�9 �7:7655� 10�13


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Fig. 11. Results obtained from Eq. (76) for the third-order non-linear gain constant K3 of the Duffing oscillator.

Fig. 12. Results obtained from Eq. (76) for the fifth-order non-linear gain constant K5 of the Duffing oscillator.


for all cases i.e. the signal should have even order components. Hilbert transformation requiresthe previous estimation of a Fourier transformation. An objective way of assessing linearity is toverify the proportional property of the linear systems. The steps taken here are to excite thesystem at two levels, X0; and 2X0; then obtain the MSE between the two responses. The input is aband-limited white noise.

The first-order amplitude is straightforward to obtain in both cases. The AFRF of the ALEs ismore difficult to isolate as the other order increases. On the one hand, if the input amplitude intothe system is too low, the higher-order response can be indistinguishable from the noise and onthe other, the magnitude of the harmonics that the non-linear systems produces scales with apower of the corresponding order of the input amplitude [1]. The relative magnitude betweenconsecutive non-linear operators is more significant as the input amplitude increases.

If the system is free of noise, small amplitudes should produce well-separated non-linear-ordercomponents, but with noise, the minimum amplitude is limited. The best results are found when

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Table 3

Non-linear gain constants of the Duffing oscillator and the amplitude ratio obtained from them

Third-order Fifth-order Seventh-order

Kn calculated 2:51� 1010 7:6843� 1010 7:8131� 1010

Kn true 25� 1010 7:5� 1010 7:5� 1010

an calculated 2:39� 109 7:3446� 109 7:4693� 109

Fig. 13. AFRF models obtained for H13ðO2Þ from the frequency spectra analysis obtained from circle fitting, the

harmonic constant K3 of the Duffing oscillator.


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harmonic constant K5 of the Duffing oscillator.


harmonic constant K2 of the electrostrictive actuator.


the MSE between 2yðxðtÞÞ and yð2xðtÞÞ is of around 5%; this corresponds to a root-mean-square(RMS) of 1070 kN for the Duffing oscillator and of 4 kV for the electrostrictive actuator.

The data obtained are the sum of all the Volterra operators (Eq. (39)). So from these data it isnecessary to estimate the individual values. The estimation is carried out in ascending orderaccording to the following formulas.First order:

y1ðtÞDyðtÞ � y0; ð77Þ

Second-order ALE:

y2ðtÞDyðtÞ � y0 � y1ðtÞ; ð78Þ

Third order:

y3ðtÞ ¼ yðtÞ � y0 � y1ðtÞ � y2ðtÞ; ð79Þ

etc. For the Duffing oscillator neglect the term containing y0 as this system does not possess anyDC term.

The evaluation of both methods can be assessed observing Figs. 13–16 where the estimatedAFRF with model fits are shown. Also, the mean square error of each model with respect to thespectra is shown in Table 4.

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harmonic constant K2 of the electrostrictive actuator.


7. Discussion and conclusions

If a non-linear system possesses convergent Volterra series representation, it can be identi-fied using the decomposition in ALEs. For this objective, two different approaches have beenused here; to find each AFRF and the non-linear gain constant Kn directly (the ratio between thenth-order AFRF and the first-order one). Each method possesses its own strengths andweaknesses.

Because all the higher-order ALEs have the same damping ratio and natural frequency, thedirect AFRF identification procedure offers a means to evaluate both parameters several times. Inthe examples presented here, four higher-order ALEs were identified, so that for each system therewere four different evaluations of the damping ratio z and the natural frequency on from differentdata groups (each group is produced for a specific order signal). As the unique different parameterbetween the ALEs is the modal constant, the non-linear gain constant offers a second evaluationof the amplitude ratio an:

For the Duffing oscillator, both methods seem to produce accurate results, very close to eachother. The electrostrictive actuator results seem to be good for the first two orders, the third- andfourth-order ALEs are a little harder to obtain. The accuracy in the case of the Duffing oscillatoris observed to be higher, this could be because of the nature of the systems. The Duffing oscillatorpossesses a ‘‘natural’’ first-order response and the same can be said for higher-order responses.The electrostrictive actuator is forced to produce a first-order output by adding the DC. Thischaracteristic makes the response of the high-order ALEs a little closer in magnitude to eachother, leaving a smaller amplitude interval to isolate a particular-order response.

To be able to apply these methods, it is necessary to have the possibility to test the system inseveral levels of excitation until the proper level for testing is obtained (Section 6).

The method’s capability of obtaining higher frequency order responses lay strongly in anaccurate first-order modelling. This is because the trial inputs for the non-linear part areconformed by products of lower order signal produced by simulation. As the order becomeshigher, the error tends to find a clean spectrum.

Appendix A. The Hammerstein model of an electrostrictive actuator

The theoretical electrostrictive actuator is based on the characteristics of the Lead-Lanthanum-Zirconium-Titanate (PLZT) electostrictive material. When the grain size is of the order of 4:5 mm;

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Table 4

Mean square error obtained from the models of the Duffing oscillator and the electrostrictive actuator

Duffing Electrostrictive

oscillator actuator

Model H13 H15 H12 H13 H14

Circle fitting 0.0315 0.3897 0.3897 0.9918 1.5678

Non-linear gain constant Kn 0.0351 0.3771 0.3771 0.6256 1.5709


its strain-electric field e–E curve in the d33 direction is shown in Fig. 17 that also contains theaverage strain for a given electric field E: Its saturation point is at a field of 22 kV=cm:

It is usual to model the electrostrictive effect using the quadratic law as

eðtÞ ¼ d33E2f ðtÞ; ðA:1Þ

where d33 is the electrostrictive constant when the electric field is applied in the same direction asthe controlled deformation. Considering the mid-points curve in Fig. 17, it is found that d33 ¼0:13 cm2=kV2: However, no second-order law is able to model the electrostrictive response up tosaturation. It is found that a sixth order is the lower-order approximation that produces a goodapproximation:

e ¼ d 033E2

f þ fE6f ðA:2Þ

by interpolation of the mid-points

e ¼ �0:92� 10�6E6f þ 0:14E2

f : ðA:3Þ

The voltage x needed to obtain a specific electric field E is x ¼ EL; where L is the length of theelectroceramic in the 3-direction.

The electrostrictive actuator system model is shown in Fig. 18 under an external excitationforce F ðtÞ:

The structural stiffness is represented by k and the electrostrictive stiffness is k0: The viscousdashpot and the stiffness of other structural elements are represented by well-knownmathematical models. The electrostrictive effect does not produce stress in the ceramic, butsimply changes the length of the ceramic element. The stress is produced because of the stiffness-strain that is obtained from the difference of the actual length of the ceramic at the time t and the

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Fig. 17. Strain–electric field response of the PLZT: the quadratic law partially models the electrostrictive effect.


position yðtÞ of the mass m at the same time. Then,

F 0 ¼ �k0ðy � DðtÞÞ; ðA:4Þ

where DðtÞ is the increment in length of the ceramic produced by the electrostrictive effect:

DðtÞ ¼ eðtÞL: ðA:5Þ

Here eðtÞ is the strain obtained from Eq. (A.1), and L is the ceramic length in the deformationdirection.

Adding all the forces in the direction of y; one obtains

Fk þ Fct þ F 0 ¼ Fi; ðA:6Þ

where Fi is the inertia force, Fk is the equivalent structural stiffness of all other elements apartfrom the electroceramic, Fc is the viscous damping force and F 0 is the force exerted by the ceramicstress–strain. When expressions for all the forces are substituted in Eq. (A.6) one obtains

m .y þ c ’y þ ðkt þ k0Þy ¼ k0Di: ðA:7Þ

The equivalent stiffness of an axially loaded member is

k0 ¼EA

L; ðA:8Þ

where E is the Young’s modulus. Substituting Eqs. (A.1), (A.5) and (A.8) into (A.7) we obtain

m .y þ c ’y þ ðkt þ k0Þy ¼EA

Ld33E2

f Li: ðA:9Þ

Finally, using (A.4) and the total system stiffness k;

m .y þ c ’y þ ky ¼EA

L2d33xðtÞ2: ðA:10Þ

Dividing by the mass, an equation that agrees with Eq. (3) is obtained:


EA

L2md33xðtÞ2: ðA:11Þ

For a more accurate model for the electrostrictive effect, the sixth-degree polynomial inEq. (A.2) is used. In the same way as for the square law, the corresponding Hammerstein model is

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Fig. 18. 2A. The Wiener model.


obtained as


k0d 033

Lmx2 þ

k0f

L5mx6: ðA:12Þ

The data used in this equation are, L ¼ 12 mm; A ¼ 0:3� 4 mm; m ¼ 4:8 g; E ¼ 100 GPa; d 033 ¼

0:14� 104 cm2=kV2 ¼ 1:4e� 15 m2=V2 and f ¼ �0:78� 10�10 cm6=kV6 ¼ �9:2� 10�41 m6=V6:In the particular case that the mass m is suspended by the electrostrictive plate, Fkt is zero andthen,

k ¼ k0 ¼EA

L¼ 107 N=m:

Then,

on ¼

ffiffiffiffik

m

r¼ 45643:54 rad=s:

The final version of the electrostrictive model is therefore,

.y þ 9128:7 ’y þ 2:0833� 109y ¼ 2:4306� 10�4x2 � 7:027� 10�22x6: ðA:13Þ

References

[1] J.A. Vazquez-Feijoo, K. Worden, R. Stanway, On perturbation analysis and the Volterra series, Proceedings of the

Seventh International Conference of Structural Dynamics; Recent Advances, University of Southampton, 2000,

pp. 415–425.

[2] S.J. Gifford, Volterra series analysis of non-linear structures, Ph.D. Thesis, University of Edingorgh, 1975.

[3] C.W. Bert, Material damping: an introductory review of mathematical models, measures and experimental

techniques, Journal of Sound and Vibration 29 (2) (1959) 129–153.

[4] J.A. Vazquez-Feijoo, K. Worden, R. Stanway, Linearization of electrostrictive-type systems, Proceedings from the

Conference on Healt Monitoring and System Identification, Madrid, Spain, 2000, pp. 115–126.

[5] I.J. Leontaritis, S.A. Billings, Input–output parametric models for non-linear systems; Part I: deterministic non-

linear systems, International Journal of Control 41 (1971) 303–328.

[6] E.J. Ewins, Modal Testing: Theory and Practice, Research Studies Press Ltd., John Willey, NY, 1970.

[7] K. Worden, G.R. Tomlinson, Nonlinearity in Structural Dynamics, Detection, Identification and Modelling,

Institute of Physics Publishing, Bristol, Philadelphia, 2001.

[8] J.S. Bendat, Nonlinear Systems Techniques and Applications, Wiley, Interscience, New York, 1998.

[9] S.A. Billings, K.M. Tsang, Spectral analysis of block-structured non-linear system, Mechanics and Systems Signal

Processing 4 (1978) 117–130.

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System Identification Using Associated Linear Equations