Application of Singular Perturbation Theoryto Hydraulic Servo Drives - System Analysis and

Control Design

Bernhard Manhartsgruber

Department of Mechanics & Machine DesignJohannes Kepler University

Altenbergerstr. 69, 4040 Linz, Austriamanhartsgruber@mechatronik.uni-linz.ac.at

This paper describes the system analysis and control design of an electrohydraulic servo-drive using anonlinear composite feedback control approach based on singular perturbation theory (Kokotovic, Khalil& O’Reilly 1986). A comparison with linear output feedback (P-control) and linear state feedback (LQR-design) shows the bene¿ts of the proposed method. The nonlinear control design features feedbacklinearization of the slow system (reduced system, corresponding to the incompressible model) and sim-ple proportional feedback of the pressure difference (fast system). Experimental results are given forstep responses and sine tracking.

Key words: hydraulic drives, nonlinear control, composite feedback, singular perturbation

1 INTRODUCTION

Electro-hydraulic control systems are used for the exertion of large forces on their surroundingsor for the exact manipulation of heavy objects. Applications include pick and place robotsfor die-cast part handling, oscillation drives for continuous casting machines, force and rollinggap control in rolling mills, positioning of aircraft control surfaces and control of military gunturrets.Simple proportional output feedback as described in (Merritt 1967) suffers from the low damp-ing of hydraulic servo-drives.State feedback techniques have successfully been implemented to improve the dynamic behav-ior of hydraulic position control systems. To keep track with variations of system and loadparameters, various adaptive control methods have been tested. Still, the behavior of hydraulicdrives is inherently nonlinear. Especially in applications where the dry friction force at the hy-draulic cylinder is high, or where the pressure drop at the servo valve becomes low comparedto the load pressure – i.e. where the drive reaches its power limits – the nonlinear effects cannotbe neglected.This paper deals with the comparison of nonlinear composite feedback design for hydraulicservo drives (Manhartsgruber & Scheidl 1998) to LQR design and conventional P-control. Thethree control designs are implemented on a test rig with a driven mass of 545 kg and a displace-ment of150 mm. The controllers are designed to handle a step input of 30 mm height withoutsaturation of the control input or the chamber pressures.In additition to the investigation of the step inputs, the problem of harmonic oscillation is ana-lyzed. Even in the case of equal piston areas and a fully symmetric design of the entire system,a strong nonlinear effect is reported for the sine tracking problem due to the nonlinear pressure-Àow relation of hydraulic valves (Feuser & Piechnick 1993).

Proportional feedback with a control gain near the stability margin results in oscillatory stepresponses in the experiments. The sine tracking performance of this type of control is low dueto nonlinear effects. While the LQR design provides suf¿cient damping and fast step responses,it does not improve the sine tracking behavior. The nonlinear composite feedback control solvesthe sine tracking problem along with a further improvement of the step response.

2 MATHEMATICAL MODEL

p , V1 1

p , V2 2

A

pS

p0

FP

x

xv

m

Fig. 1: Hydraulic servo drive.

Figure 1 shows a servo-drive with a hydraulic cylinder featuring equal areas and a directioncontrol valve. The cylinder acts on the heavy mass6. The process force8� is an unknowndisturbance. The supply pressureR7 and the return line pressureRf are assumed to be constant.This can be accomplished by the use of hydraulic accumulators near the control valve.

2.1 Equation of Motion with Dry Friction

Due to the tight sealing, hydraulic cylinders feature a strong dry friction effect. The frictionforce is denoted by8so. The behavior of this friction force is rather complex. The frictionforce not only depends on the piston velocity, but also on the pressure in the cylinder chambers.Additionally, the sealing system has internal dynamics: The deformation of the seal rings due tothe pressurization of the sealing system takes some time. In the case of very small displacementsthe contact force of the seal rings is dominated by viscoelastic reaction forces caused by thesmall deformation. Therefore, the dry friction model is not valid for small displacements. In thiscase, a critical value of approximately 20-40>m can be observed in experiments. Furthermore,the friction behavior is subject to changes due to temperature variations, aging and wear.To the authors knowledge, the friction behavior of hydraulic cylinders under dynamic con-ditions, i. e. the aforementioned internal dynamics of the sealing system, has not yet beeninvestigated. Experimental studies under stationary conditions (Gessat 1997) show a typicaldry friction behavior: The friction force decreases with increasing velocity. Furthermore, thefriction force increases with increasing chamber pressure. In this paper a model of the form

8so ' r�}?

�_%

_|

�3EC_ff n _f�e

3k

������_%

_|

������4FD �

�� n &

�R� � R7

2

�n &

�R2 � R7

2

��

is used to cope with the inÀuence of the chamber pressures R� E|� and R2 E|�. The equilibriumstate of the drive with 8� ' f is % ' _%*_| ' f and R� ' R2 ' R7*2. Thus, the dependenceof 8so on R� and R2 is linearized at R� ' R2 ' R7*2. A signi¿cant increase of the friction forceat higher velocities (up to 0.3 m/s) did not appear in the experiments. Thus, a linear term in thefriction force model as used in (Yun & Cho 1991) is missing. The equation of motion reads

6_2%

_|2' ER� � R2��� 8so � 8� .

2.2 Pressure Dynamics

The bulk modulus of oil varies with temperature and pressure (Jinghong, Zhaoneng & Yuanzhang1994, Manring 1997). In the case of small temperature variations and moderate supply pressurevalues, a constant bulk modulus q can be assumed. The evolution of the chamber pressures isthen described by

T�fn�%q

_R�_|

' ��_%_|

n ^T ER7 � R��� ^T ER� � Rf�

nr}�

%T%Tc4@

�'?

tR73R�{R?

� r}�� %T

%Tc4@

�'?

tR�3Rf{R?

T2f3�%q

_R�_|

' �_%_|

n ^T ER7 � R2�� ^T ER2 � Rf�

nr}�� %T

%Tc4@

�'?

tR73R2{R?

� r}�

%T%Tc4@

�'?

tR23Rf{R?

where T�f and T2f are the initial chamber volumes at % ' f and '? denotes the nominal Àowrate of the valve at a nominal pressure loss {R? per notch. The valve opening %T is masked bythe function

r} E%� G'

;?=

f % f% e,re� % : �

.

In addition to the square root behavior of the ori¿ce Àow, there are always small leakage Àowsat the control valve. The leakage Àow is proportional to the pressure drop with a coef¿cient ^T .Since the leakage is inÀuenced by the valve spool position, this model is only an approximationof the real behavior. Yet, the effect of leakage for the closed valve (%T ' f) is precisely incor-porated in the mathematical model. Therefore the leakage model is a good approximation forsmall valve openings. For large valve openings the inÀuence of the leakageÀow is negligible.

2.3 Valve Dynamics

If the effects of hysteresis andÀow force are neglected, the dynamics of the valve can bedescribed by an LTI system. The transfer function can be estimated from experimental in-put/output data using Welch’s periodogram method. Figure 2 shows the results of this non-parametric identi¿cation method together with a 6th order transfer function

-40

-30

-20

-10

0

10

mag

nitu

de [

dB]

10 100 1000

-500

-400

-300

-200

-100

0ph

ase

[°]

frequency [Hz]

Fig. 2: Estimated transfer function of the servo valve and least squares ¿t.

C Er� '@f

@f n @�rn @2r2 n @�r� n @ere n @DrD n rS

computed by a least squares ¿t. The coef¿cients @� are given in table 1.Due to saturation effects, the linear behavior is only valid for very small amplitudes. Thefrequency analysis was performaed with input/output data of a logarithmic sine sweep between1 Hz and 10 kHz with a demand amplitude of 1 % of the full valve opening.

2.4 Dimensionless Equations

The application of perturbation techniques requires a properly scaled formulation of the prob-lem. This is achieved by a dimensionless time coordinate

� ' / |uE � '

_

_�'

�

/

_

_|and by rescaling of the dependent variables in the form

�� E�� 'R� E|�

R7�2 E�� '

R2 E|�

R71 E|� '

% E|�

%4@ l E|� '

%T E|�

%Tc4@ ,

with the supply pressureR7, a rated displacement%4@ , and the maximum valve opening%Tc4@ .With the new parameters

Tf 'T�f n T2f

2e '

T�f � T2fT�f n T2f

@/ ' %Tc4@ '?

/Tf

uR7{R?

c K '�%4@

TfS/ '

R7 �

6%4@ /2c 0 '

R7qc

_/ff '_ff

6%4@ /2c _/f� '

_f�6%4@ /2

c ^/ '^T R7/Tf

the model may be written as

d3 5=588:5< � 4355d4 5=567677 � 434<d5 9=6763;< � 4348d6 4=3;9869 � 4345d7 4=:8;4<7 � 43;d8 4=43<;6: � 437

Table 1: Coef¿cients of the transfer function C Er�.

0 E� n en K 1� ��� ' �K �1 n ^/ E�� 2��� n r} El� @/s�� �� � r} E�l� @/

s�� (1a)

0 E�� e� K 1� ��2 ' K �1 n ^/ E�� 2�2� n r} E�l� @/s�� �2 � r} El� @/

s�2 (1b)

�1 ' S/ E�� � �2�� s� � (1c)

r�}?��1��

_/ff n _/f�e3km �1m� � E� n & E�� n �2 � ���

3 EXPERIMENTAL SETUP

The test rig with a driven mass of 6 ' DeD kg is shown in ¿gure 3. The other parameters canbe found in table 2.The experiments were performed with a MOOG D760-995A type servo-valve. The nominalÀow rate at a pressure drop of�D bar per notch is2� l/min at full opening. Due to a signi¿cantsaturation of theÀow curve, theÀow gain is higher for small openings. Since the valve isoperated in the linear range during the experiments, the saturation is not modelled.The dimensionless parameters related to the scaling frequency/ ' � rad/s are

@/ ' ��H2c K ' f��bc S/ ' .H�Dc 0 ' f�ff.D

_/ff ' 2��Sc _/f� ' ���Hc k ' 2�f e ' �f��Dc ^/ ' f�fSe .

4 LINEAR CONTROL

In the following linear analysis the valve dynamics is assumed to be in¿nitely fast. Thus, thecontrol input� is equal to the valve openingl. The dry friction effect must be neglected inorder to make the system linearizable. With the state transformation

'

5997

j{1�1

6::8 '

5997

�� n �2 n e E�� � �2�� ��� � �2

1�1

6::8 ,

the linearization at the resting point1 ' �1 ' j ' { ' f with input l ' f yields

� '

59999997

�2^/0

2 e^/0

f f

2e

�� e2^/0

�2� n e2

�� e2^/0

f �2�

�� e2K

0f f f �

f S/ f f

6::::::8 n

5999997

fs2

�� e2@/0

f

f

6:::::8 l .

Fig. 3: Experimental Setup.

Classical feedback design (Merritt 1967) uses linearized models at midstroke (e ' f), becausethe stability margins are worst at this position. With e ' f, the model is

� '

5999997

�2^/0

f f f

f �2^/0

f �2K

0f f f �

f S/ f f

6:::::8 n

5999997

fs2@/0

f

f

6:::::8 l .

The ¿rst state variable j becomes nonobservable and noncontrollable. The stability of thissubspace is guaranteed by ^/ : f. Other authors (Bobrow & Lum 1996, Plummer & Vaughan1996) omit this analysis of the dynamics of the pressure sum and de¿ne the net force on thepiston or the pressure difference as a state variable and work with a state vector like

3 '�{ 1 �1

�Aresulting in a third order model. This reduction of the state space results in

�3 '

597 �2

^/0

f �2K

0f f �S/ f f

6:8 3n

597

s2@/0

ff

6:8 l .

4.1 Proportional Feedback

Most industrial controllers for hydraulic drives use proportional feedback with a demand input1oes in the form

l ' &��1oes � 1

�.

Symbol Value Unit RemarkssV 433 bar supply pressure� 46633 bar effective bulk modulusD 9> 74 cm5 effective piston area{pd{ 483 mm rated displacementY43 758 cm6 volume of left chamberY53 8:8 cm6 volume of right chamberg33 4<3 N friction parametersg34 <8 N� 5 1

Table 2: Physical Parameters.

The closed loop model is

�3 '

597 �2

^/0

�&�

s2@/0

�2K

0f f �S/ f f

6:8 3n

597 &�

s2@/0

ff

6:8 1oes

with a stability criterion

f &� 2s2K ^/0 @/

� ��2 .

A gain of &� ' ��f is used in the experiments. With this gain the controlled system handlesinput steps of �f mm without saturation (cavitation or pressures above the supply pressure) ofthe chamber pressures. The LQR controller and the nonlinear controller are designed to meetthe same restrictions for a step height of �f mm.The proportional gain &� is limited by the amount of leakage at the control valve. Alternatively,the damping can be modelled by internal leakage at the cylinder or by viscous friction forces.Yet, the modelling of leakage at the valve is more accurate and realistic, since the sealing of thecylinder chambers is almost ideal and viscous friction forces can be neglected at low velocities.

4.2 LQR Control

In order to improve the dynamical behavior of the controlled drive, a state feedback with anextra demand input

l ' &��1oes � 1

�� &( �1 � &{{

can be designed by either pole placement (Finney, de Pennington, Bloor & Gill 1985, Vaughan& Plummer 1990, Plummer & Vaughan 1996) or by optimization of a performance index(Bobrow & Lum 1996, Jacobs & Roth 1982). The performance index used is

"]f

�3A"3n o l2

�.

This is the well known LQR-problem (Athans & Falb 1966). The model

�3 '

57 �f��. f �Df�.

f f �.H�D f f

68 3n

57 .2f

ff

68 l

with the weight matrices

" '

57 f�f� f f

f �f ff f f

68 o ' �

yields the controller gains

&� ' ���Sc &( ' f�fS�c &{ ' f��D .

5 NONLINEAR COMPOSITE FEEDBACK CONTROL

The linearization of the system for the design of linear controllers results in the paradox situationthat the leakage of the valve limits the controller gain. Most of the dissipation in the systemis due to dry friction effects. Thus, a nonlinear control design should yield better results. Themethod of composite feedback control (Kokotovic et al. 1986) uses a two time-scale approachfor the design of slow and fast feedback functions for the slow and fast dynamics of the system.The application of this model for the position control of hydraulic servo systems has beenproposed in (Manhartsgruber & Scheidl 1998).The model (1a,1b,1c) is rewritten in the new state

3 '

59997

P

{

1�1

6:::8 '

59997

�� n �2

�� � �2

1�1

6:::8

yielding

0 E� n en K 1��Pn �{2

' �K �1 n ^/ E�� P�{� (2a)

n r} El� @/

t�� Pn{

2� r} E�l� @/

tPn{2

0 E�� e� K 1��P3 �{2

' K �1 n ^/ E�� Pn{� n (2b)

n r} E�l� @/

t�� P3{

2� r} El� @/

tP3{2

�1 ' S/{� s� � r�}?��1�� (2c)�

_/ff n _/f�e3km �1m� E� n & EP� ���

The dynamic behavior of the servo-valve is described in the frequency domain by1

l ' C Er� � .

5.1 Exact Feedback Linearization of the Reduced Model

The servo-valve is very fast compared to the dynamics of the ”slow” states1 and �1. Therefore,the valve dynamics is neglected (l ' �) in the reduced model. Furthermore, the unknowndisturbance forces� E�� is neglected in the following control design.Setting0 ' f in the differential equation system (2a,2b,2c) results in a differential algebraic

4 The signals � and x are denoted without hats (a�> ax) in the frequencey domain, since the hat denotes fast timein this paper.

system where the algebraic subsystem has the unique solution

Pf ' �,

{f ' �@/2^/

v�2

@2/e^2/

n r�}?��1��12K

^/n 2� r�}?

��1��2

@2/e^/

� K �1

^/. (3)

Substitution of these solutions into eq. (2b) yields

�1 ' S/{f � s� � r�}?��1��

_/ff n _/f�e3km �1m� . (4)

Thus, the reduced system is of second order and has the control input �. The goal of the exactfeedback linearization is to ¿nd a slow feedback function

� ' Kr

�1c �1c �

�that results in the standard linear behavior

�1 n 2 B /S�1 n /2

S1 ' /2S� (5)

of the drive position 1 due to the new control input �. A comparison of the nonlinear reducedmodel (4) with the linear equation (5) results in the requirement

{f

�1c �1c �

�'

�

S/

�/2S E� � 1�� 2B/S

�1 n r�}?��1��

_/ff n _/f�e3km �1m�� . (6)

This can be achieved by the inversion of equation (3) resulting in the slow feedback function

Kr

�1c �1c �

�'

K �1 n ^/{f

�1c �1c �

�@/

yxxw 2

�� r�}?��1�{f

�1c �1c �

�with {f

�1c �1c �

�according to eq. (6).

10-3 10-2 10-1 100 101 102 103

-100

-50

0

50

mag

nitu

de [

dB]

10-3 10-2 10-1 100 101 102 103-720

-540

-360

-180

0

pha

se [°

]

ω [1]

Fig. 4: Bode Plot of the Open Loop Transfer Function.

-ψ

1

ψ2

z

z

−1

NumericalDifferentiation

s

s sc

c c

ωω δω

2

2 2 2 1+ +Reference Model Friction Force

ξfrF

Slow Feedback

vs ,,ξξζ

k∆-

∆

ζs

ζf

ξv

ξ

ζ

Hydraulic Drive

( ).

.( )

.

Fig. 5: Composite Feedback Control with Reference Model for Friction Compensation.

5.2 The linearized dynamics of the fast variables

The fast dynamics is investigated in the fast time scale

� '�

0c E �� '

Y

Y�' 0

Y

Y�.

The transfer function of the valve dynamics may be rewritten by

T Er� '@f

@f n @�0 rn @2 E0 r�2 n @� E0 r�

� n @e E0 r�e n @D E0 r�

D n @S E0 r�S

where @� ' @�*0�. The Laplace variable r is transformed into the new time by

r ' 0 r .

The valve transfer function in the transformed frequency domain is

T Er� '@f

@f n @�rn @2r2 n @�r� n @ere n @DrD n @SrS.

In the fast time scale the linearized dynamics of P and { at 1 ' �1 ' f is

�P�

{�

�'

597 �2

^/�� e2

2^/ e

�� e2

2^/ e

�� e2�2

^/�� e2

6:8 �

�P{

�n

597 �s

2@/e

�� e2

s2@/

�

�� e2

6:8 l .

The transfer function from the input l to the output { is

C Er� '

s2@/

�� e2rn 2^/�

rn2^/� n e

��rn

2^/�� e

� .

The inÀuence of the reference position e is negligible. The nominal model at midstroke (e ' f)

is

Cf Er� '

s2@/

rn 2^/.

5.3 Proportional Control of the Fast Dynamics

The slow feedback function Kr copes with the dominant nonlinearities. The deviations of theperturbed system behavior from the reduced model – i.e. the compressibility effects – have to beeliminated by a second feedback, the fast feedback functionKs . Since the deviations betweenthe perturbed equations and the reduced system are� E0� small (Fenichel 1979) a simple linearcontrol design can be used for this purpose.In this paper, a proportional feedback

Ks

�1c �1c �c{

�' &{

�{f

�1c �1c �

��{

�with the desired pressure difference according to eq. (6) and a constant gain&{ is designed inthe frequency domain. Figure 4 shows the magnitude and phase plots of the open loop transferfunction&{ � T Er� � Cf Er� for &{ ' �. The stability margin is reached at&{ ' e�2. A rangeof &{ ' f�.D to &{ ' ��D has been successfully used in the epxeriments to get a well dampedresponse of the pressure control.

5.4 Composite Feedback

The control loop is closed by simply adding the slow and fast feedback functions to a compositefeedback (Kokotovic et al. 1986)

� ' Kr

�1c �1c �

�n Ks

�1c �1c �c{

�.

The structure of the control system is shown in¿gure 5. The velocity is not measured butcomputed by numerical differentiation of the position signal. The friction force compensationcan be improved by using the velocity of a reference model according to eq. (5) instead of thesignal from the numeric differentiation.The parameters used in the following experiments are

/S ' 2Z � e�Dc B ' f�Hc &{ ' f�.D .

6 EXPERIMENTAL RESULTS, CONCLUSIONS, AND OUTLOOK

6.1 Step responses

The step responses for the P-controlled system are given in¿gure 6a. Step heights of 30, 15 and6 mm have been tested according to a step from�f��c�f�fDc�f�f2 tonf��cnf�fDcnf�f2 andback in the normalized scale. Since the scaling frequency is/ ' �, the time scale� coincideswith the physical time| measured in seconds.The step response behavior for LQR control in¿gure 6b is much better, with higher usage of theavailable valve opening. The feedback of the pressure difference{ results in a small increaseof the stationary position error due to the stiction effect. The usual workaround for this problemis an addtional feedback of the integrated error. In the presence of dry friction such additionalintegrators must be disabled for small position errors to avoid limit cycles.

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

ξ , ξ

ref

0 0.5 1 1.5 2

-2

-1

0

1

2

dξ /

dτ

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

∆

0 0.5 1 1.5 2-0.2

-0.1

0

0.1

0.2

ζ

τ

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

ξ , ξ

ref

0 0.5 1 1.5 2

-2

0

2

dξ /

dτ

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

∆

0 0.5 1 1.5 2-0.5

0

0.5

ζ

τ

0 0.5 1 1.5 2

-0.1

-0.05

0

0.05

0.1

ξ , ξ

ref

0 0.5 1 1.5 2

-2

0

2

dξ /

dτ

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

∆

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

ζ

τ

a) b) c)

Fig. 6: Step responses: (a) P-control, (b) LQR-control, (c) nonlinear composite feedback control

The nonlinear composite feedback control (Figure 6c) shows a further improvement of the stepresponse. The behavior of the nonlinear control for small position errors will be improvedin future work by adding the aforementioned error integrator together with a small region ofinactivity around zero. Furthermore, the behavior of friction of hydraulic cylinders should beinvestigated in more detail.

6.2 Sine oscillation

The bene¿ts of nonlinear control can be shown by a comparison of the sine response of thehydraulic drive. The sine oscillation is best viewed in a phase space plot with position 1, velocity�1 and acceleration �1. The acceleration is replaced by a measurement of the force in the pistonrod. The rod force 8 was normalized in the way

s '8

R7�.

In addition to the three-dimensional1 � �1 � s trajectories, a projection onto the�1 � s -plane isshown in¿gure 7.The systems with proportional feedback (Figure 7a) and LQR control (Figure 7b) show a non-linear behavior of the trajectories during harmonic oscillations. The nonlinearities can be com-pensated by nonlinear control (Figure 7c) to give a precise harmonic motion of the mass. Thesine oscillations were performed at a frequency of� Hz with Amplitudes of 150 and 105 mm(1 and 0.7 in the dimensionless scale). The trajectory with 150 mm amplitude is slightly af-fected by the nonlinearity of theÀow curve of the servo-valve. The problem could be solved byincorporating this nonlinearity in the model.

-5

0

5

-1.5 -1 -0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4

0.6

dξ/dτ

ξ

f

-5

0

5

-1.5 -1 -0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4

0.6

dξ/dτξ

f

-5

0

5

-1.5 -1 -0.5 0 0.5 1

-0.4

-0.2

0

0.2

0.4

0.6

dξ/dτξ

f

a) b) c)

Fig. 7: Sine oscillation: (a) P-control, (b) LQR-control, (c) nonlinear composite feedback control

7 NOMENCLATURE6 mass of moving parts [kg] ^T leakage param. [mes/kg]� effective piston area [m2, cm2] / time scaling factor [1/s]q effective bulk modulus [N/m2, bar] | time [s]%4@ rated displacement [m, mm] % E|� position [m]T�fc T2f initial chamber volumes [m�, cm�] R� E|� c R2 E|� pressures [N/m2, bar]R7 supply pressure [N/m2, bar] %T E|� valve opening [V]Rf return line pressure [N/m2, bar] @/ Àow gain [1]{R? nominal pressure drop [N/m2, bar] K displacement ratio [1]'? nominal Àow rate[m�/s, l/min] S/ inertia force ratio [1]_ff kinetic Coulomb friction [N] ^/ leakage constant [1]_ff n _f� static Coulomb friction [N] � dimensionless time [1]k friction force decay exponent [s/m] �� E�� c �2 E�� dimensionless pressure [1]& sensitivity: friction/pressure [m2/N] 1 E�� dimensionless position [1]8so friction force [N] l E�� valve opening [1]%Tc4@ maximum valve opening [V]

REFERENCES

Athans, M. & Falb, P. (1966). Optimal Control, McGraw-Hill.Bobrow, J. E. & Lum, K. (1996). Adaptive, high bandwith control of a hydraulic actuator,

Transactions of the ASME. Journal of Dynamic Systems, Measurement, and Control118: 714–720.

Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations,Journal of Differential Equations 31: 53–98.

Feuser, A. & Piechnick, M. (1993). Antriebsdimensionierung unter berücksichtigung vonzylinderÀüchen- und steuerkantenverhältnissen,O+P Ölhydraulik u. Pneumatik37: 509–516.

Finney, J. M., de Pennington, A., Bloor, M. S. & Gill, G. S. (1985). A pole assignmentcontroller for an electrohydraulic cylinder drive,Transactions of the ASME. Journal ofDynamic Systems, Measurement, and Control 107: 145–150.

Gessat, J. (1997). Reibungsverhalten von Hydraulikdichtungen und Führungselementen,O+PÖlhydraulik und Pneumatik41: 743–746.

Jacobs, M. & Roth, J. (1982). Rechnerunterstützte Auslegung und Realisierung von Regelungs-konzepten für elektrohydraulische Antriebe,5. Aachener Fluidtechnisches Kolloquium,Fachgebiet Hydraulik, Band 2, pp. 239–262.

Jinghong, Y., Zhaoneng, C. & Yuanzhang, L. (1994). The variation of oil effective bulk modu-lus with pressure in hydraulic systems,Journal of Dynamic Systems, Measurement, andControl 116: 146–150.

Kokotovic, P. V., Khalil, H. K. & O’Reilly, J. (1986).Singular Perturbation Methods in Con-trol: Analysis and Design., Academic Press, London etc.

Manhartsgruber, B. & Scheidl, R. (1998). Non-linear control of hydraulic servo-drives basedon a singular perturbation approach,in C. A. Burrows & K. A. Edge (eds),Proceedingsof Bath Workshop on Power Transmission and Motion Control, Professional EngineeringPublishing, London, pp. 301–313.

Manring, N. D. (1997). The effectiveÀuid bulk-modulus within a hydrostatic transmission,Journal of Dynamic Systems, Measurement and Control 119: 462–466.

Merritt, H. E. (1967).Hydraulic Control Systems, John Wiley, New York.Plummer, A. R. & Vaughan, N. D. (1996). Robust adaptive control for hydraulic servosys-

tems,Transactions of the ASME. Journal of Dynamic Systems, Measurement, and Con-trol 118: 237–244.

Vaughan, N. D. & Plummer, A. R. (1990). Some issues in pole placement control of electro-hydraulic servos,Proceedings of Third Bath International Fluid Power Workshop, Bath,UK, pp. 6–28.

Yun, J. S. & Cho, H. S. (1991). Application of an adaptive model following control tech-nique to a hydraulic servo systeme subjected to unknown disturbances,ASME Journal ofDynamic Systems, Measurement and Control 113: 479–486.