978-1-4244-8452-2/11/$26.00 ©2011 IEEE FPM 2011

Investigation of Speed Control of a Pressure Coupling Energy Saving Hydraulic System

Kyoung Kwan Ahn, Triet Hung Ho and Hoang Thinh Do School of Mechanical Engineering

University of Ulsan Ulsan, Republic of Korean

kkahn@ulsan.ac.kr

Abstract— In this paper, an Adaptive Fuzzy Sliding Mode Control (AFSMC) was proposed for speed control of a hydraulic pressure coupling drive. The AFSMC combined a direct adaptive fuzzy scheme and a fuzzy sliding scheme in a new structure to reduce the tracking error and the chattering of the control effort. Experiments were performed with different controllers, the AFSMC, the traditional Sliding Mode Control (SMC) and the PID controllers, and different of operating conditions. Then, the experimental results were brought into comparison to evaluate the effectiveness of the AFSMC controller from the view points of stability, performance and robustness of the closed loop system.

Keywords— hydraulic systems, secondary control systems, pressure coupling systems, adaptive fuzzy sliding mode control

I. INTRODUCTION Hydraulic pressure coupling systems have been considered

energy-saving systems because of their energy recovery potential. However, they have been rarely employed in industry as well as mobile applications. One of the most obstacles was the difficulty in speed or position control of secondary units which were variable displacement hydraulic pump/motors. In such systems, speed of a secondary unit must be controlled by a feedback controller and it was controlled by directly adjusting its displacement [1].

In hydraulic pressure coupling systems, each secondary unit was considered as a controlled plant in which the control input was the electric signal into its displacement control mechanism (DCM) and the control output was its speed or position. The operating pressure and the fluid flow were considered as the power source for secondary units. In addition, the mechanical and the volumetric losses of the secondary unit were varied based on its operating conditions and they were often nonlinear. Therefore, the control system was a nonlinear system with model uncertainty.

Robust control was employed for the speed control of such the systems as follows. Robust H2 controller was used in [1]. The use of load observer robust controller with or without consideration of saturation of the displacement was shown in [2-3]. Bilinear optimal controller was studied in [4]. The stability of the closed system was proven and the results of these studies showed that good performances were achieved within small variation of the operating pressure. However, in some applications such as a hydraulic hybrid vehicle, the operating pressure was varied so much to satisfy the

requirements of representative speed profiles [5-7]. The uncertainties of the system were always substantial due to external loads, wind effort, friction of roads. Thus, another control scheme should be employed to guarantee the stability and the performance of the closed system for above cases.

In this paper, we firstly propose the use of AFSMC for speed control of a hydraulic pressure coupling drive. For a secondary unit, the controlled plant, it is difficult to achieve an exact model but the governing form of the controlled plant can be analytically obtained. Then experiments were performed with different controllers and the experimental results were brought into comparison to evaluate the effectiveness of the AFSMC.

II. BRIEF DICRISPTION OF A HYDRAULIC PRESSURE COUPLING DRIVE SYSTEM

2.1 Principle of working Schematic diagram of a hydraulic pressure coupling drive

with one secondary unit is shown in Fig. 1. It is a hydrostatic drive which transmits the energy from the supply part to the load part. The supply part can be an electric motor for industrial applications or a combustion engine for mobile applications. The load part can be winch systems or a vehicle and its chassis.

M

LPHP

PM2P1

ps

pr

q1 q2

qa

�2, T2�1

u1, D1 u2, D2

Load

part

FL

Hydrostatic driveEnergy

supply

part

EM

LS

LR

Fig. 1 Schematic of hydraulic pressure coupling drive A novel characteristic of the drive is that the secondary unit

can work as a hydraulic pump or hydraulic motor to drive the load or recover the kinetic energy of the load. To drive the load, the secondary unit functions as a hydraulic motor, the hydraulic energy is transmitted to the kinetic of the load. To decelerate the load, PM2 functions as a hydraulic pump to generate the braking torque. Thus, the kinetic energy of the load is recovered and stored in the high pressure accumulator HP.

691

2.2 Modeling of main components in the system

2.2.1 Hydraulic pump/motor The ideal volumetric flow rate of fluid through a hydraulic

machine Qi and the ideal torque at the shaft of the machine Ti are expressed by Eq. (1) and Eq. (2), respectively.

maxiQ D��� (1)

maxiT pD�� � (2) where, � is the displacement ratio, the fraction between current and the maximum displacement of the machine. Dmax, � and �p = ps -pr are the maximum displacement, speed and pressure difference between two ports of the machine, respectively.

2.2.1.1 Pump mode The pump volumetric efficiency is defined as the ratio of the

actual flow rate Qa to the ideal flow rate Qi, as expressed by Eq. (3),

1QQa loss

vP DQ maxi� ��� � � (3)

The torque or mechanical efficiency of a hydraulic pump which is a relationship between the actual torque Ta and the ideal value Ti. It is estimated by Eq. (4),

max

1

1i i

tPlossa i loss

T TTT T T

D p

�

�

� � �� �

�

(4)

2.2.1.2 Motor mode The volumetric and the mechanical efficiencies of the

pump/motor in the motor mode are expressed as follows,

max

1

1i i

vMlossa i loss

Q QQQ Q QD

�

� �

� � �� �

, (5)

max

1a i loss losstM

i i

T T T TT T D p

��

�� � � �

�. (6)

From above equations, Tloss and Qloss are the torque and the volumetric losses of a hydraulic axial machine. Generally, values of Tloss and Qloss vary depending on the load conditions and experimental determination [37].

2.2.1.3 Electro-hydraulic displacement control mechanism For variable displacement hydraulic pump/motor, the

displacement of the machine is usually controlled by an electro-hydraulic mechanism. Although the mathematical model of the mechanism is a fifth-order system, a reduced-order model is usually used in practical applications[1-3]. A pure linear first-order system [2-3] is used in this study as follows:

1

sv sv

u tK K� � �� �

1K

� ��� (7)

where u(t) is the electric signal control, � is the time constant, Ksv is the DC gain of the DCM.

2.2.2 Hydraulic accumulator The structure and parameters of the accumulator can be

founded in [6]. The gas pressure in the accumulator is assumed to be equal to the operating pressure in the hydrostatic drive. The gas pressure is expressed by the following equation for

our purpose although the accumulator was also more exactly modeled [6].

0 0n n

ga ga ga gap V p V� (8)where, Vga Vga0 are the current and the initial gas volume, pga and Vga0 are the current and initial pressure in the accumulator, n is the polytropic exponent of gas within the range from 1 to 1.4. The gas volume in the accumulator is estimated by Eq. (9).

0 1 2ga gaV V q q dt� � �� (9)From equations 8 and 9, we see that the operating pressure

is controlled by controlling the pump flow rate q1. The operating pressure can generate hydraulic torque at the secondary unit’s shaft. The hydraulic torque is either Ta = Ti-Tloss or Ta = Ti+Tloss when the secondary unit works in the motor or the pump modes, respectively.

2.2.3 The load In this study, the load is a simple flywheel. To model the load, the most importance is the model of friction of the load. The simplest one is Ffr = C�. More accuracy and continuous, the friction model is expressed in Eq. (10) [41].

.exp / .fr c s cT T T T a C sgn� � � �� � � � �� � (10) where, Tc is the Coulomb friction and (Ts-Tc).exp(-|�|/a) is the the Stribeck friction. And the coefficient a is the reference speed approximately within the range from 0.001m/s to 0.01m/s. The dynamic equation of the flywheel is obtained by applying Newton’s second law, as in Eq. (11),

a fr exT J T T�� � �fr exT Tfr efrTTffr , (11) where, Tex is the braking torque J is flywheel moment of initial.

2.3 Modeling of the controlled plant For speed control of the flywheel the controlled plant

includes the secondary unit and the load. The input of the controlled plant is the electric signal to its DCM u2 and the output is the speed of the flywheel �. Block diagram of the controlled plant is presented in Fig. 2a and 2b.

PT1 P

Dmax

u2

�D2

ps

� Ti

I��

Look up table

Torque losses

Ta

Friction

�� �

Tloss

Tfr

-

Tex

Displacement control mechanism

Secondary Unit PM2

Load

Fig. 2a Block diagram of the controlled plant

Secondary unit with input dead-zone

PT1 P

Dmax

u2 �D2

ps

� I

Torque lossesdepending on speed

+ Stribeck friction

Ti

Viscous friction

�� �

Tloss

Tfr

-

P

C

Tex

Linearization model of Load

Unknown Disturbance

w

Fig. 2b Block diagram of the controlled plant with input dead-zone

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The controlled plant is expressed in Eq. (12) max. .

1 sv

p DC dJ J

K w

y

�� �

� �� �

�

�� � � � ���� � � ���� ���

CJ

� �� � ��

1� ��

� � �� (12)

Where, w is the output of the input dead-zone mechanism, d is unknown disturbance. The disturbance in this case includes the Stribeck friction, the torque losses of PM2 which depends on the speed and even the external brake torque. We see that there is no direct relationship between the output y and the control input w. To generate a direct relation between w and y, we differentiate y. The result is the first equation in 12, yy is still not directly related to w, we differentiate it again. Thus, the controlled plant is a two relative degree system, n = 2. After some manipulation, we obtain, , .y f g w D� �� � �y f f . (13) where f(�,�) is a unknown function of (�, �) because C, J and � are not exactly known, g = g(�p, �) is not included state variables (�, �) of the plant. D stands for all uncertainties and external disturbances. In Eq. (13), w is not physical control input. Thus, we re-expressed Eq. (13) in term of the physical control input u2. For simplicity of presentation in the speed controller synthesis u is used instead of u2.

The dead-zone with input u and output w is expressed by Eq. (14).

.

0

.

p p

n p

n n

m u b for u b

w u for b u b

m u b for u b

� � ���� � � � ��� � ���

(14)

Characteristics of the dead-zone are that w is not measureable, bn<0, bp>0, m >0; bp ,pn and m are bounded or

,min ,max,p p pb p p ��� � , ,min ,max,n n nb p p ��� � and � �min max,m m m� . We can re-write the model of w as expressed in Eq. (15)

.w m u u � � . (15) Where m is called general slope of the dead-zone, (u) can be calculated as

.

.

.

p p

n p

n n

m b for u bu m u for b u b

m b for u b

� ���

� � � ���� ��

. (16)

Here, m, pp, bn and u !� are bounded. max ,max max ,minmax ,p nm p m p! � � , pn,min carries a negative value.

Applying for the controlled plant, the controlled system (13) can be re-written as follows,

, . .y f g u g u D� � "� � � �y f f . (17a) Where, g’ = g.m. From Eq. (17a), the term of g. (u)+D stands for uncertainties and disturbance of the controlled plant.

Note that the control input u is not included into the controlled plant when .u m u � � . Thus, the controlled plant can be expressed by Eq. (17b)

, . 'y f g u D� � "� � �y f f (17b) where D’ includes the disturbances D and the model uncertainties bp, bn and m.

III. ADAPTIVE FUZZY SLIDING MODE CONTROLIn this paper, an adaptive fuzzy sliding mode including a

direct adaptive fuzzy controller and fuzzy sliding mode based controller. For the controlled plant 17a, if f, g, and D’ were known, we could design the following controller for speed control of the secondary unit.

1

*

0

1 , ' sgnn

i ni d

iu f c e y D s

g� � �

�

�

# $� � � � � �% &" ' () . (18)

Equation 18 is rewritten as follows * * *

eq ru u u� � , (19a)

1

*

1

1 , 'n

i neq i d

iu f c e y D

g� �

�

�

# $� � � � �% &" ' () , (19b)

* sgnru sg�

�"

, (19c)

where, *equ and *

ru are the ideal equivalent and the reaching controllers, respectively, and sgn is the sign function. � is a positive constant standing for uncertainties of the model of the controlled plant [43]. Where s is the sliding surface and it is defined as

1 21 2 1 1

n nns e c e c e c e e c e� ��� � � � � � �1 1c e1 11 �c e ee1122�c e22 �e . (20)

where, e = y-yd, The coefficients ci, i =1,…n-1 are chosen so that the polynomial (20) is Hurwitz. The designed controller is satisfied the condition,

212

d s sdt

�� � . (21)

Then, the system is stability and lim 0t e*+ � [43] or limt dy y*+ � . However, f, g’, and D’ are not known, so the ideal control

law 18 is not implemented. Moreover, the sign function in the reaching controller 19c generates the chattering phenomenon in practice. Here, the use of a direct adaptive fuzzy controller is proposed to mimic the equivalent controller and a fuzzy controller is used in the reaching controller.

3.1 Direct adaptive fuzzy equivalent controller (DAFEC) To express the equivalent control law 19b by a fuzzy logic

system, we employ a point fuzzification method, product-type inference and center-average defuzzifier. The fuzzy rule base consists of collections of fuzzy IF-THEN rule as follows.

R(i): IF � is 1iL and T is 2

iL THEN ufz = ,i Where T is measured torque at the shaft of the secondary

unit. 1iL and 2

iL are linguistic values of the speed and the torque of the controlled plant, respectively.

2 2

1 21 2

1 2

exp , expj ji im T mL L T

��

- -

� �� �# $ # $. / . /� � � �% & % &. / . /' ( ' (� � � �

(22)

where, mkj and -kj are the center and the width of the membership functions of the speed and the torque, k = 1,2: two inputs, j = 1,…n numbers of membership functions of each input. The output of the fuzzy system is expressed as follows,

1

11

ˆˆ ,M

Mi ii

fz i iMiii

Xu f X X

X

, 0, , 1

0�

��

� � �) ))

(23)

693

Where X = (x1, x2)T = (�, T)T, M is the numbers of rules, ,i is the output of the fuzzy system due to rule R(i). 2

1i

i j jjX L x0

��2 , i X1 � 0 and 1

1Mii

X1�

�) .

Let 1 2, ,..., TM, , , ,� be the adaptive parameter vector

and 1 2, ,... MX1 1 1 1� be the regressive vector of the adaptive fuzzy controller. According to the universal approximation theorem [30], there exist an optimal fuzzy control system

* *,fzu X , in the form of (23) such that

* * * *1

, Meq fz i ii

u u X X, 3 , 1�

� � �) . (24) Where, 3 is the approximation error and it is assumed to be bounded by E3 � with E is a positive constant. In the fact that, the approximated adaptive fuzzy controller ˆ fzu is employed instead of the optimal one * *,fzu X , . Thus, existing of an error defined as follows,

* * *ˆ ˆ , ,fz fz eq fz fzu u u u X u X, , 3� � � � �ˆfz fzu u uˆfz fzu uuf . (25) The arm of the adaptive fuzzy synthesis is that to find out

the adaptive law to force the state s and *, , ,� � *, , , *,, tend to zero.

3.2 Fuzzy reaching controller (FRC) If the reaching control law 19c is employed the chattering

phenomenon is occurred. There are several solutions for that problem [43-47]. In this paper, a fuzzy logic system is used to reduce the effect of the sign function in the reaching control law 19c. The use of the fuzzy logic system reduces the chattering phenomenon but keeps the tracking error small. Moreover, the determination of the upper bound of the reaching controller is not easy so a bound estimation mechanism is employed. As a result, the reaching control law is expressed by Eq. (26).

,ˆ ˆˆ , .sgnr r fzu s E E s u� (26)

Where E is the bound estimation which determines the approximated error of the adaptive fuzzy control law and it is a positive value. ,0 1r fzu� � is a normalized fuzzy controller which is designed as follows.

The input of the reaching fuzzy control is the value of s and its output is the value of ,r fzu . The controller ,r fzu is designed based on the following rules. The magnitude of ,r fzu increases when s increases and vice versa. The fuzzy logic system of the fuzzy reaching controller is shown in Fig. 3a. In the IF part of the reaching fuzzy system, the triangular membership functions are employed while the singleton-type are used for the THEN part. The numbers of rules and values of parameters in the reaching fuzzy system are selected by trial and error. In this paper, 5 rules are used as follows,

R(1) IF |s| is Z (zero) THEN is ,r fzu Z R(2) IF |s| is PS (positive small) THEN ,r fzu is PS R(3) IF |s| is PM (positive middle) THEN ,r fzu is PM R(4) IF |s| is PNB (near positive big) THEN ,r fzu is PNB R(5) IF |s| is PB (positive big) THEN ,r fzu is PB The defuzzifier of the FRC is used the center-gravity

method.

9, 1

.r fz k kku s0 �

��) (27)

Where, 0k(s) is output value of the membership function of the IF part of the RFLC .

|s/ �4

,r fzu

10

10 �5 �6 �7 �8

PBZ PNBPMPS

�9

Fig. 3a Membership functions of IF and THEN parts of the FRC

Here, � is the width of the boundary of s, and the value of � is selected by trial and error. From the rules base of the FRC if s is outside of the boundary, the reaching controller is the same with the control law (19c). If value of s is small or inside the boundary the reaching controller is a nonlinear function of s as in (27). Figure 3b shows a equivalent input – output relationships of the fuzzy reaching controller , .sgnr fzu s with � = [0.25; 0.6; 0.8; 0.92; 1] and a sat function. When compare to the sat function, the FRC has a stronger affect for small values of |s/�|. Value of the output of the fuzzy controller is always positive and the minimum value can be designed by the fuzzy rules. The chattering phenomenon is reduced because of the adaption mechanism of the boundary

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Out

put

Input (s/�

sat (s/�) u

r,fzsgn(s)

Fig. 3b the equivalent input-output relationships of the fuzzy reaching

controller , .sgnr fzu s

3.3 The adaptive fuzzy sliding mode control (AFSMC) and its stability analysis

Given the system (17b) with the AFSMC controller ˆˆ ˆ ˆ, ,fz r

u u X u s E,� � (28) with the control laws expressed in equations (23), (26) and

(27) and the adaptation laws including dead zone modification in equation (29)

1. .s X, � 1�� 1, � 11. .s.�� (29a)

2 , 0

0

.ˆ0

r fzs u if sE

if s

�� � ��� �� ���

2E�� 2 s����� ��� (29b)

where, �0 is a designed positive constant parameter. Then the closed loop system is stability from Lyapunov’s sense and the tracking error converges to zero asymptotically.

Finally, the structure of the closed loop system is shown in Fig. 3c. The adaptive fuzzy sliding mode control comprises the

694

sum of the direct adaptive fuzzy controller ˆ fzu and the fuzzy reaching controller ˆru .

Slidingsurface

Adaption law

Fuzzy reachingcontrol controller

Controlled plant

Equivalent fuzzycontroller

�r�e

dedt

es

-

+

+

-

,

E

Adaption law

T

AFSMC

u

ˆ fzu

ˆru,ˆr fzu

Fig. 3c Structure of the AFSMC

IV EXPERIMENTS

4.1 Test bench setup Schematic of the test bench are shown in Fig. 4a. In the test

bench, hydraulic pump/motors with maximum of 55cc/rev are employed. The moment of inertia of the flywheel is 2.5 kgm2. The volume and the gas pre-charge pressure of the high accumulator HP are 20L and 120bar, respectively. The setting pressure of the relief valve is 200bar.

Control and Data Acquisition system are performed by a personal computer and 12 bit A/D and D/A boards 1720 and 1711 of Advantech products. The control system is designed in Matlab Simulink with the sampling time is 2ms.

�A/DBoard

Power supply

Computer

M

ps

pr

Flywheel FL

Hydraulic

brakeLP

HP

P1 PM2

CV

RL

u2u1u3

pr

ps

q

D/A board

T

Fig. 4a Schematic diagram of the test bench

Figure 4b shows the photograph of the hydraulic part in the test bench. The hydraulic part is separated into two parts, the hydraulic supply part, in the right hand side of the figure, and the actuator part in the left.

E. MP1Brake

LP

HP

PM2

FL

SS

Fig. 4b Photograph of the hydraulic part

The brake in the test bench only has a function of generating the external braking torque. The value of the braking torque is controlled by control the proportional reducing valve. The torque and speed sensors are marked SS in Fig. 4b. Two accumulators HP and LP are placed near PM2 to guarantee the working condition of the secondary unit.

4.2 Experimental results In this paper, the speed control of the secondary unit is

focused only. Thus, the pump P1 and the brake are simply controlled. The pump P1 is ON (u1 = 1) or OFF (u1 = 0) based on the operating pressure. The pump P1 is ON until ps reaches 150bar then it is OFF until ps decreases to 125bar. Here, u3 is used to control the braking torque and only an open loop control of u3, the setting value of u3, is satisfied the purpose.

To evaluate the effectiveness of the AFSMC for speed control of the secondary unit, four different speed references are employed including 10-mode profile, multiple step, sine and triangle references with three control schemes, the traditional SMC, the PID control and the AFSMC. The 10-mode profile is one of Japanese schedules for vehicle testing. The PID controller in the paper is designed by trial and error where KP = 0.0045V/rpm, KI = 0.0005 V/rpm and KD = 0.00009V/rpm. In addition, the designed PID controller is able to be employed in four different speed references. Therefore, the tracking errors are not exhaustedly eliminated in the experimental results by the designed PID controller. SMC controllers using sat function were introduced to reduce the chattering phenomenon. In this paper, the AFSMC controller is brought into comparison with a SMC controller using a sat function for the reaching controller and the PID controller. To design the SMC controller equations 19 are employed except that a sat function is used instead of a sign function. Parameter used for designing the SMC controller is obtained from the nominal model of the system and the value of � is determined as shown in [43]. The nominal model is obtained by using Eq. (12) and Eq. (14) with �p = 140 bar, �= 0.3 s, J = 2.5 kg.m2, C = 0.05 Nm.s/rad, Dmax = 55 cc/rad and Ksv = 55.10-6/3.4: m3/rad/V.

Initial value of , is randomly selected in [-1, 1]. We select, c1 = 3.75 �1 = 16 and �2 = 0.2 for the experiment. The input of the FRC is s/�, the value of � is selected based on the control system. If value of � is small ur get tend to the sign function as in 19c and � is large ur is smooth. Five membership functions are employed for the IF part of the FRC as shown in Fig. 3a. The parameters of the THEN part are given as, � = [0.25; 0.6; 0.8; 0.92; 1].

The experimental results are shown in Fig 5. The figures show the results of the system in case of no external braking torque generated by the brake and the operating pressure varies from 125 bar to 150 bar. The experimental results indicate that the tracking errors of PM2 with the AFSMC are significantly reduced when compared with the PID and the SMC. The tracking errors are significantly reduced by using the AFSMC controller due to the effects of both the fuzzy adaptive controller and the fuzzy reaching controller. Note that the traditional PID or SMC controller used in the figures 5a

695

and 5b is able to achieve tighter tracking by redesigning the controller’s parameters. Figure 6 shows the experimental results of the speed control of PM2 for different operating conditions. To evaluate the robustness of the speed controller, two values of external braking torques, the 50 Nm and the 20 Nm, are applied. Figure 6c shows the response of the system for multiple-step with a 50 Nm external torque which is applied at 35th second and then it is released at 50th second. The results in Fig. 6 imply that the controller is robust with external step disturbance.

20 40 60 80 100 120 140

0

500

1000

1500

Spe

ed R

PM

Proposed PID SMC

20 40 60 80 100 120 140

-20

0

20

40

60

Err

or R

PM Proposed

PID SMC

Time secondsFig. 5 Response of the system for 10-mode profile

20 40 60 80 100 120 140

0

500

Spe

ed R

PM

Proposed PID SMC

20 40 60 80 100 120 140-20

0

20

40

Err

or R

PM

Proposed PID SMC

Time seconds Fig. 5b Response of the system for triangle reference

20 40 60 80 100 120 140

0

400

800

1200 50 Nm 20 Nm

Spe

ed R

PM

20 40 60 80 100 120 140-40

-20

0

20

40

Err

or R

PM

20Nm 50Nm

Time Seconds Fig. 6a Response of the system for 10-mode profile with external loads

20 40 60 80 100 120 140

0

500

Spe

ed R

PM

50 Nm 20 Nm

20 40 60 80 100 120 140-40

-20

0

20

40

Err

or R

PM

20 Nm 50 Nm

Time seconds Fig. 6b Response of the system for triangle reference with external loads

V. CONCLUSIONS In this paper, a hydraulic pressure coupling system was

briefly presented from the operating principle to the governing dynamic equations. The input dead-zone characteristic of the controlled plant was taken into account during stability analysis of the closed loop system.

An AFSMC was proposed for speed control of a secondary unit of a hydraulic pressure coupling system. The proposed controller comprised two parts, the direct adaptive fuzzy equivalent controller and the fuzzy reaching controller. Here, the measured torque at the shaft of the secondary unit was employed as a second fuzzy input of the adaptive fuzzy equivalent controller. The closed loop system was analytically proven the stability with Lyapunov’s stability theorem.

The experimental results confirmed the effectiveness of the proposed controller from the speed tracking performance to the robustness for different speed references and different operating conditions. The tracking errors of the proposed controller were always smaller than those of a traditional SMC using sat function and a PID controller for all speed references in this paper.

REFERENCES

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[3] C.S. Kim, C.O. Lee, Robust Speed Control of a Variable-Displacement Hydraulic Motor Considering Saturation Nonlinearity, ASME, Vol. 122 , 2000, pp. 196-201.

[4] L. Guo, H. Schwarz, “A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive”, Proc. of 28th Conference on Decision and control, Tampa, Florida, 1989, pp. 542-547.

[5] R. Johri, Z. Filipi, “Low-cost pathway to ultra efficiency car: Series hydraulic hybrid system with optimized supervisory control”, SAE, 2009-24-0065.

[6] P. Matheson, J. Stecki, “Modeling and simulation of a fuzzy logic controller for a hydraulic hybrid powertrain for use in heavy commercial vehicles”, SAE, 2003-01-3275.

[7] Y.J. Kim, Z. Filipi, “Simulation study of a series hydraulic hybrid propulsion system for a light truck”, SAE, 2007-01-4151.

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