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Synthesis and implementation of non-integer integrators using RLCdevicesRoy Abi Zeid Daou ab; Clovis Francis c; Xavier Moreau a

a Bordeaux I, LAPS, Univeriste de Bordeaix 1, Cours de la liberation - Bat. A4, Bordeaux, France b

Faculty of Sciences and Engineering, Department of Engineering, Holy Spirit University, Kaslik,Jounieh, Lebanon c Department of Electrical Engineering, Lebanese University, Tripoli, Lebanon

Online publication date: 01 December 2009

To cite this Article Daou, Roy Abi Zeid, Francis, Clovis and Moreau, Xavier(2009) 'Synthesis and implementation of non-integer integrators using RLC devices', International Journal of Electronics, 96: 12, 1207 — 1223To link to this Article: DOI: 10.1080/00207210903061980URL: http://dx.doi.org/10.1080/00207210903061980

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Synthesis and implementation of non-integer integrators using RLC

devices

Roy Abi Zeid Daoua,c, Clovis Francisb and Xavier Moreaua

aBordeaux I, LAPS, Univeriste de Bordeaix 1, Cours de la liberation - Bat. A4, Bordeaux,France; bDepartment of Electrical Engineering, Lebanese University, El Kobbe, Arz street,

Tripoli, Lebanon; cFaculty of Sciences and Engineering, Department of Engineering, Holy SpiritUniversity, Kaslik, Jounieh, Lebanon

(Received 20 October 2008; final version received 4 April 2009)

The aim of this article is to introduce a new method used to implement non-integeroperators, which is based on R, L and C electrical devices. Several configurationsof RLC devices are emphasised. Our results show that only specific arrangementsof RLC devices may lead to non-integer behaviour. We present a new synthesismethod which is used to determine the electrical devices value based on high levelparameters. We investigate a particular case of identical RLC cells which is widelyused in automotive suspension controllers’ implementation. Our results areapplied to the implementation of the Citroen active suspension developed earlierby the Laps department (Moreau, Altet and Oustaloup 2004, ‘The CRONESuspension: Management of Comfort-Road Holding Dilemma’, Journal ofNonlinear Dynamics, 38, 467–484) in collaboration with the PSA Company.

Keywords: non-integer integration; phase constancy; fractional controllers; robustcontrol; hydropneumatic suspension

1. Introduction

A great importance has been recently accorded to non-integer operators,such as integrators or differentiators (Oldham and Spanier 1974; Miller andRoss 1993; Samko, Kilbas and Marichev 1993). In fact these operators werewidely used in different fields of engineering, such as the fractional controllersdesign (Oustaloup 1995), modelling of several physical phenomena like thethermal diffusion (Cois 2002), electromagnetic waves propagation (Poinot andTrigeassou 2004), etc.

Electrical resistive and capacitive (RC) cells were used to implement non-integerintegrators (Podlubny, Petras, Vinagre, O’Leary, and Dorcak 2002) since electricaldevices are popular and are adequate for electrical simulations. Using the BondGraph approach (Dauphin-Tanguy 2000), implementation results obtained with RCcells can be extrapolated to other engineering domains like the hydropneumatic fieldwhich is used to implement non-integer operators used in automotive applications(Moreau, Ramus-Serment and Oustaloup 2002). However, RC cells present a majorlimitation since they are unable to take into consideration the inductive effect

*Corresponding author. Email: cfrancis@ul.edu.lb

International Journal of Electronics

Vol. 96, No. 12, December 2009, 1207–1223

ISSN 0020-7217 print/ISSN 1362-3060 online

� 2009 Taylor & Francis

DOI: 10.1080/00207210903061980

http://www.informaworld.com

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observed when long connections are used between RC hydropneumatic cells. Figure1 presents the two RC arrangements used in the hydropneumatic domain in order tobuild the hydropneumatic suspension (Serrier, Moreau and Oustaloup 2007). Theinductive effect becomes dominant for long hydropneumatic wires as will be shownby the results presented in this article (case of wires longer than 1 m).

This article is composed as follows: In section 2, we will introduce the differentRLC arrangements used to implement non-integer operators with the related inputimpedance (Zin(s)). In section 3, we present the synthesis method used to determinethe component’s values by imposing the high-level parameters proposed byOustaloup (Oustaloup and Bansard 1993; Oustaloup 1995; Oustaloup, Levron,Mathieu, and Nanot 2000). These high-level parameters are the low- and high-cutofffrequencies, the derivation order and the constant gain K that provides a unit gain,0 dB at the unit angular frequency, 1 rad/s. In section 4, we consider the case ofidentical cells, which is commonly used in industrial applications. By the end insection 5, we consider the case study of the Citroen car hydropneumatic suspensionin an attempt to apply our results to this domain.

2. RLC arrangements

In this section, we present the different configurations of RLC cells to be used in thenon-integer integrators implementation. We present also the synthesis method used tocalculate the different component values in such a way that the transfer function ofthese circuits describes a non-integer behaviour. We have to distinguish between theway the cells are connected and the way the devices are interconnected in each cell.

2.1. Parallel arrangement of RLC cells in series

Figure 2 presents the first configuration we have studied, which is based on theparallel connection of a number of cells where each cell is composed of RLCelements connected in series.

Figure 1. Hydropneumatic arrangement of RC cells.

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The input impedance of this arrangement is given by

ZeðsÞ ¼VinðsÞIinðsÞ

¼ 1

C0sþPN

i¼1Cis

LiCis2þRiCisþ1ð1Þ

where N is the number of cells of the arrangement.

2.2. Cascade configuration of RLC cells in gamma

Three different configurations can be found in this case as presented in Figure 3.These configurations are obtained by changing the position of the different deviceswith respect to each other inside an RLC cell. Note that the components used in thiscase are ideal (lossless) and the time response of the circuit to a given input signal isobtained by a simple time domain convolution of the input signal with the circuit’simpulse response. In order to take into consideration the non-linearities of thecomponents, we have to determine the circuit’s Volterra kernel (Serrier, Moreau,Sabatier and Oustaloup 2006b), which is outside the scope of this article.

The input impedance related to each of the Gamma configurations can be writtenas:

Gamma1 configuration:

Zin sð Þ ¼1

C0 s þ 1

L1s þ R1 þ1

C1 s þ1

L2s þ R2 þ 11

::::1

CN�1 s þ 1

LNs þ RN þ1

CN s

ð2Þ

Figure 2. Parallel arrangement of RLC cells in series.

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Gamma2 configuration:

Zin sð Þ

¼ 1

C0sþ 1

R1þ 11

L1sþ 1C1s

þ 1

R2þ 11

L2sþ 1C2s

þ::::þRN�1þ11

LN�1sþ 1CN�1s

þ 1

RNþLNsþ 1CNs

ð3ÞGamma3 configuration:

Zin sð Þ

¼ 1

C0sþ 1

L1sþ1

1

R1þ1

C1s

þ 1

L2sþ1

1

R2þ1

C2s

þ::::þLN�1sþ 11

RN�1þ1

CN�1s

þ 1

RNþLNsþ1

CNs

ð4Þ

Figure 3. The three different Gamma configurations (a): Gamma1, (b): Gamma2, (c):Gamma3.

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2.3. ZY approach

Adding or deleting cells in the case of the Gamma configurations implies re-calculating the whole circuit transfer function. To solve this problem, we introducethe ZY approach which deals with each cell as independent from the rest of thecircuit. After calculating the impedance of all independent cells, we compute theglobal circuit’s impedance. So, the addition or the elimination of a cell or a group ofcells does not require the recalculation of the relations 2, 3 or 4 in order to find thenew transfer function.

Figure 4 presents a system with Nþ 1 cells. Each cell is composed of animpedance Zi and admittance Yi. These two elements can be the combination ofcapacitive, resistive and inductive components. F(s) and E(s) are related to theLaplace transforms of flux, f(t), and effort, e(t), in the Bond-Graph theory. Thesymbols ‘in’ and ‘out’ correspond, respectively, to the system’s input and output(Podlubny et al. 2002).

In order to facilitate the computation of the transfer function of thisarrangement, a matrix approach is adopted. Thus, for a cell of rank i consideredseparately, the input/output relation of the power variables Fi(s) and Ei(s) is

Ein sð ÞFin sð Þ

� �i

¼ Ti½ �Eout sð ÞFout sð Þ

� �i

ð5Þ

where [Ti] is the transfer function matrix of cell i whose expression is given by

Ti½ � ¼1þ Yi sð Þ Zi sð Þ Zi sð Þ

Yi sð Þ 1

� �ð6Þ

The input impedance (Zin(s))i of cell i is given by

Zin sð Þð Þi ¼Ein sð ÞFin sð Þ

� �i

����Foutð Þi ¼ 0

¼ 1þ Yi sð Þ Zi sð ÞYi sð Þ

ð7Þ

The global system’s transfer matrix, [T], is obtained by the product of Nþ 1 localmatrices, [Ti]

T½ � ¼ a11 sð Þ a12 sð Þa21 sð Þ a22 sð Þ

� �ð8Þ

Figure 4. ZY arrangement representation.

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From the last equation, it is easy to extract the input impedance of the globalcircuit

Zin sð Þ ¼ Ein sð ÞFin sð Þ

����Fout ¼ 0

¼ a11 sð Þa21 sð Þ ð9Þ

For all Gamma arrangements, we use the approach described in this paragraphand we affect the following values to the impedances and admittances

Z0 ¼ 0

Y0ðsÞ ¼ C0s

�) for all arrangements

ZiðsÞ ¼ Ri þLis

YiðsÞ ¼ Cis

�) for Gamma1 ;

ZiðsÞ ¼ Ri

YiðsÞ ¼ Cis=ðLiCis2þ 1Þ

�) for Gamma2 ;

ZiðsÞ ¼ Lis

YiðsÞ ¼ Cis=ðRiCisþ 1Þ

�) for Gamma3

Equation (6) leads to calculate the transfer function matrix of each cell. Bymultiplying the resulting matrixes we obtain the matrix T (Equation (8)) and by theend, we apply Equation (9) to compute the input impedance of the whole circuitwhich is equivalent to the expressions represented in the relations 2, 3 and 4 whenusing the Gamma1, Gamma2 or Gamma3 arrangements, respectively.

3. Fractional integrator synthesis

The transfer function of the ideal fractional integrator is given by

HðsÞ ¼ 1

s1�mð10Þ

where m is the differentiator’s order (0�m� 1). In other words, the Bode plot of theideal integrator shows a magnitude drop by 20 (1 7 m) in dB/decade and a constantphase equal 90 (m7 1) in [8] all over the frequency axis.

Since the integrator is band-limited in the frequency domain between thefrequencies ob and oh, Equation (10) can be written in the form (Trigeassou, Poinot,Lin, Oustaloup and Levron 1999; Lin 2001; Cois 2002)

ImðsÞ ¼K

s

1þ sob

1þ soh

!m

ð11Þ

According to Oustaloup’s rational approach (Oustaloup and Bansard 1993;Oustaloup 1995; Oustaloup et al. 2000), relation (11) can be approximated by arecursive distribution of N poles and zeros

INðsÞ ¼K

s

YNi¼1

1þ so0

i

1þ soi

!ð12Þ

where oi and o0i are the poles and zeros of the new transfer function.

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Figure 5 shows the asymptotic Bode diagram for both the band-limitedfractional derivator (Dm(s)) (the solid line) and the rational approach derivator(DN(s)) (the dotted-dashed line). We can see that, using both approaches, thefractional behaviour is present between the cut-off frequencies ob and oh.

The synthesis algorithm is based on the following steps:

(1) Define the high level parameters that specify the fractionalintegrator behaviour:ob: low frequency of the fractional integrator,oh: high frequency of the fractional integrator,m: derivation order,K: a constant that provides a unit gain at the frequency om (usually equals to1 rad/s).

(2) Define the number of RLC cells which determines the number of poles andzeros in Equation (12).

(3) From the above data, determine the value of the recursive factors a and Z bymeans of the following formulas

aZ ¼ oh

ob

� �1N

; Z ¼ aZð Þ1�m; a ¼ aZð Þm ð13Þ

(4) The value of K is calculated so that the magnitude of the integrator I(jo) isequal to the unit at the frequency om ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiob:ohp

Figure 5. Asymptotic Bode diagram for Dm(s) and DN(s), and hence for Im(s) and IN(s), withm2 ]0,1[.

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(5) Calculate the N zeros and N poles of IN(s) (without taking into account thepole s¼ 0) using the following recursive relations:

o10 ¼ ffiffiffiffi

Zp

ob; oN ¼1ffiffiffiZp oh;

oi

oi0 ¼ a > 1

o0iþ1oi¼ Z > 1;

o0

iþ1oi0 ¼

oiþ1

oi¼ aZ > 1

ð14Þ

(6) After calculating the poles and zeros, we realise the fractional integrator bycalculating the N resistors, the N inductors and the Nþ1 capacitors (the firstcell is composed of a capacitor only; this capacitor has a dominant effect onthe phase constancy with a reduced number of RLC cells (Ramus-Serment,Moreau, Nouillant, Oustaloup and Levron 2002)).

Riþ1Ri¼ 1

aLiþ 1Li

¼ 1a2:Z

Ciþ 1Ci

¼ 1Z

8>>>><>>>>:

ð15Þ

Note that the first inductance value is obtained by the oscillation pulsation of thefirst pole w0.

o0 ¼1ffiffiffiffiffiffiffiffiffiffiffiL1C1

p ð16Þ

The phase constancy for the parallel arrangement is observed when the twofollowing conditions are satisfied for each cell:

Damping ratio condition :2xo0¼ R:C ) x ¼ R

2

ffiffiffiffiC

L

r> 1 ð17Þ

Ratio factor : RC =ðL=RÞ >> 1 ð18Þ

Example: Consider the case of a fractional integrator of order m¼ 0.35 with a lowfrequency ob of 10

73 rad/s and a high frequency oh of 103 rad/s. Moreover, we will

use a set of 10 cells based on resistive, inductive and capacitive components for theimplementation of this integrator. The N zeros and N poles corresponding to thisfractional integrator in Equation (12) (without taking into account the pole s¼ 0) arelisted in Table 1.

We note that between two consecutive poles, a zero is placed. This distribution ofzeros and poles ensures the fractional behaviour of the transfer function between thefrequencies ob and oh.

Table 2 presents the values of the RLC devices in each cell with the damping ratioand the value of the ratio factor (RC/(L/R)). As shown in Table 2, all damping ratiosare larger than 1 and the factor (RC/(L/R)) for all cells is greater than 100.Therefore, the two necessary criteria to obtain phase constancy hold.

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A more powerful electrical realisation of fractional integrators could be done inthe discrete domain (z plane) or in the pseudo frequency domain (w domain). Forthis purpose the RLC devices values are calculated as previously described in theanalogue domain, then the synthesised transfer function is switched to z domain orto w domain by means of appropriate transformations. The digital transfer functionis then realised by means of powerful digital signal processors (Pollock 2009) whereno limitations are imposed neither on the number of cells nor on the component’svalues.

Figure 6 shows the Bode diagram of the rational transfer function (see Equation(12)) and the synthesised integrator based on RLC cells when using the parallelarrangement of RLC cells in series. It is clear that the behaviour of the rationaltransfer function and the RLC circuit transfer function are almost the same. Thephase constancy appears between 5 6 1072 rad/s and 5 6 102 rad/s and its value isalmost 58.58 (that is equal to 90 (m 7 1) in [8]). It is clear that at unit frequency, themagnitude is almost 0 dB thanks to the effect of the constant K.

4. Fractional integrator realisation based on identical RLC cells

Identical cells play an important role in industrial applications where non-integerbehaviour is required (Serrier et al. 2007). The main advantage of using identical cells

Table 2. Values of the resistances, capacities and inductances of the controller, the dampingratio and the ratio factor for each cell.

Capacities (F) Resistances (O) Inductances (H) Damping ratio(RC/(L/R))

(6102)

C0¼ 8.91256 1072

C1¼ 5.0217 R1¼ 1.27106 102 L1¼ 9.79756 1075 1.43886 104 9.0967C2¼ 2.4551 R2¼ 6.52556 101 L2¼ 956846 1075 5.22646 103 5.24826 102

C3¼ 1.4473 R3¼ 2.77626 101 L3¼ 8.80146 1075 1.78006 103 2.78956 102

C4¼ 8.82446 1071 R4¼ 1.14266 101 L4¼ 8.75746 1075 5.73466 102 1.65266 102

C5¼ 5.42356 1071 R5¼ 4.6668 L5¼ 7.94826 1075 1.92756 102 1.11026 102

C6¼ 3.33946 1071 R6¼ 1.9031 L6¼ 7.37316 1075 6.40396 101 7.35596 101

C7¼ 2.05516 1071 R7¼ 7.76616 1071 L7¼ 5.22596 1075 2.43516 101 6.39026 101

C8¼ 1.25926 1071 R8¼ 3.18346 1071 L8¼ 2.71456 1075 1.08416 101 7.57106 101

C9¼ 7.56786 1072 R9¼ 1.33046 1071 L9¼ 2.52336 1075 3.6428 4.99026 101

C10¼ 4.13436 1072 R10¼ 6.11676 1072 L10¼ 1.72966 1075 1.4953 4.37556 101

Table 1. Values of the N zeros and N poles.

Zeros Poles

o10 ¼ 1.5667510706 1073 o1¼ 2.5409727056 1073

o20 ¼ 6.2418097146 1073 o2¼ 1.0134786626 1072

o30 ¼ 2.4888915146 1072 o3¼ 4.0393965646 1072

o40 ¼ 9.9182360426 1072 o4¼ 1.6092584466 1071

o50 ¼ 3.9509224696 1071 o5¼ 6.4093864136 1071

o60 ¼ 1.573480199 o6¼ 2.552313351

o70 ¼ 6.265585054 o7¼ 1.0162638016 101

o80 ¼ 2.4947296976 101 o8¼ 4.0462352986 101

o90 ¼ 9.9325701536 101 o9¼ 1.6109374636 102

o100 ¼ 3.954441546 102 o10¼ 6.413508386 102

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are that phase locking is observed with a very small number of cells (Serrier et al.2007). An important issue in this study focuses on the choice of the value of C0. Thisdevice is responsible for obtaining the phase locking using a reduced number of RLCcells. The drawbacks of using identical cells are the narrowband phase locking (overa single decade only) and the derivation order is between 0.5 and 0.6. This valuetends towards 0.5 for an infinite number of cells (Oustaloup 1995).

We consider first the case of the parallel arrangement of RLC cells in series. Theinput impedance of this arrangement can be re-written in the case of identical cells as:

ZinðsÞ ¼1

s C0 þNCð Þ :LCs2 þ RCsþ 1

1þ C0

C0þNC

: LCs2 þ RCsð Þ

ð19Þ

where N represents the number of identical RLC cells.This arrangement does not present a phase locking while using identical cells and

hence cannot be used to realise non-integer operators.In order to obtain a non-integer behaviour, we have to use the cascade

arrangement of RLC cells in gamma. For this reason, we studied experimentallywhether all three configurations preserve the phase locking when using identicalRLC cells.

Figure 6. Comparison of Bode diagrams for the rational transfer function and the transferfunction based on RLC cells.

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In Figure 7, which represents the Bode diagram of the Gamma1 configuration offour identical RLC cells, we have varied the value of a¼C/C0 between 1 and 10. Wecan see clearly that phase locking is observed when a¼ 6. An identical result isobserved when the Gamma2 configuration is used. However, the Gamma3configuration does not present any phase locking and hence cannot be used fornon-integer operator implementation.

Figure 8 represents the Bode diagram of identical cells for the first gammaarrangement for three values of C. In each iteration, we’ve added one cell. A weakphase locking is observed in case (a) where C¼C0. A strong phase locking isobserved in the third case (c) where C¼ 66C0. However, the phase locking is absentin the second case (b) where C0¼ 0. This result is also observed in the case of theGamma2 configuration. We remark also that phase locking becomes available whenthe number of cells reaches 4.

We deduce from the results presented in Figure 7 that the system’s behaviour, inall the three cases, is the same for low frequencies (capacitive behaviour) and centerfrequency (asymptotic performance of order 0.5). However at high frequencies, thethree circuits behave differently: while in the first and the third cases we have aresistive behaviour, in the second case we have a capacitive behaviour.

Note that the results for the Gamma3 arrangement were almost expected as thisarrangement can be described as the parallel arrangement of RLC cells in series. Infact, when the value of the inductance is too low by comparison to the values of theRC components, the circuit behaves as a parallel arrangement of RC cells in series.The phase locking for this arrangement when using identical cells is missing and thusthe fractional behaviour is absent.

5. Application

As an application of the above study, we emphasise the realisation of thehydropneumatic Citroen suspension (Altet, Nouillant, Moreau and Oustaloup2003) by means of identical RLC cells. This suspension is based on a mechanical andhydropneumatic system defined by a fractional order force–displacement transferfunction (Moreau, Altet and Oustaloup 2006). The hydraulic components used inthis technology are hydraulic accumulators (capacitive components, C, which use oiland gas separated by an impermeable diaphragm to play the same role as amechanical spring); hydraulic resistors (dissipative components, R) (Serrier et al.2006b); and hydropneumatic inductance (dissipative components, L). Figure 9represents the hydropneumatic suspension for a car’s wheel. We can see here the RCcomponents. As the wires are too short between these two components, the inductiveeffect is negligible. However, this effect is present when considering the wires thatlink the different RC cells of the car’s suspension.

The hydropneumatic RC cell configurations used to implement this suspensionare presented in Figure 1. When introducing the inductive parameter, two differentarrangements are obtained which correspond to the Gamma1 and Gamma3arrangements that we have studied in section 2 (see Figure 10). Note that theresults of the first arrangement show that the phase constancy was present whereas inthe Gamma3 configuration the non-integer behaviour was missing.

The relations that allow computing the values of the resistive, capacitive andinductive components are presented in the following equations. Note that the valueof the first capacitive element will be one sixth of the value of the other capacities.

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Figure

7.

Bodediagram

fortheinputim

pedance

usingfouridenticalRLC

cellswitha¼C/C

0.

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Figure

8.

Bodediagram

fortheinputim

pedance

fortheGamma1configuration:(a)C¼C0,(b)C¼0,(c)C¼66

C0.

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Hydraulic inductance of a channel : Lh ¼ rl

Sð20Þ

withr : Densityl : Channel lengthS : Channel Section

(

Hydropneumatic capacity : C ¼ S2vP0V0

gðMgÞ2¼ P0V0

gP2v

¼ S2v

kð21Þ

with

P0 : Inflation pressureV0 : Sphere volumeg : Constant of perfect gassesM: Suspended massSv : Jack� suspension sectiong : Gravity acceleration

8>>>>>><>>>>>>:

Hydraulic resistance : R ¼ b

S2v

ð22Þ

Figure 9. Hydraulic suspension representation (Serrier, Moreau and Oustaloup 2006a).

Figure 10. (a) First hydropneumatic suspension arrangement and its electrical equivalentcircuit. (b) Second hydropneumatic suspension arrangement and its electrical equivalentcircuit.

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Figure

11.

(a)First

hydropneumaticarrangem

entBodediagram,(b)Secondhydropneumaticarrangem

entBodediagram.

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with b: Coefficient of viscous friction

Case of the Citroen car suspension:

Inductive element calculation:l¼ 1 mr¼ 850 kg/m3.S¼ p/2 6 D2 with D¼ 1072 mThis leads to Lh� 107 Ns2/m5

RC elements calculations:P0¼ 40 barV0¼ 400 cm3

g¼ 1.4Sv¼ 3 cm2

M¼ 300 kg.

This leads to R¼ 1.6761010 Ns/m5 and C¼ 1.143610711 m5/N.The damping ratio and the factor RC/(L/R) becomes:

x ¼ 8:9

RC=ðL=RÞ � 320

The two criteria required for phase locking are met. Figure 11a,b shows the Bodediagram for both hydropneumatic configurations.

The first hydropneumatic arrangement presents a phase locking over one decadeand a derivation order around 0.58. This value tends to 0.5 when the number of cellsbecomes very important. Also, we note that the behaviour of the RC and RLCcircuits is roughly the same.

The same result holds for the second hydropneumatic configuration onceimplemented using RC cells only as can be observed in Figure 11b. However, thenon-integer behaviour is totally absent once the inductive components are taken intoconsideration.

6. Conclusion

In this article, we studied the effect of the inductive element on the non-integerbehaviour of RC cells. We’ve presented the different possible arrangementsand showed the input impedance transfer function of the RLC configurationsthat preserve the non-integer behaviour. We introduced also a new method for thesynthesis and realisation of the fractional integrator based on the RLC cells anddepicted two conditions to verify in order to obtain a non-integer behaviour.

Next, we considered the case of identical RLC cells which is widely used in thehydropneumatic domain. We demonstrated that the parallel arrangement ofidentical RLC cells in series presents no phase locking while for the Gamma1 andGamma2 configurations, phase locking is evident. We emphasised the effect of thevalue of the input capacitor C0 to get the phase constancy.

In the last part of this article, we applied our results to the Citroen car’ssuspension. For this reason, we examined the different arrangements that could be

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used in the hydropneumatic domain and showed that the results found in theprevious paragraph can be directly applied in this domain.

References

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