Abstract— This paper presents a nonlinear control approach to the position tracking for a hydraulic actuation system, which consists mostly of a servovalve and an asymmetric single-rode cylinder. The proposed control approach is derived using a sliding mode control strategy. The Tracking performance and the robustness against considerable parametric uncertainties (uncertain fluid compressibility, load variations, changes in friction constants, etc.) of the presented control method are demonstrated by both simulations and experiments under various operating conditions.

I. INTRODUCTION

YDRAULIC actuation systems are widely used in industrial applications due to among other aspects their

ability for generating large actuation forces and torques at fast motion. Apart from the highly nonlinear dynamics characteristics of hydraulic systems [1], it is also very difficult, even infeasible to mathematically collect all dynamical effects resulting (at operation) from the interactions of hydraulic components [2] (thereunder pumps, pipelines, control valves such as servovalves or proportional valves, actuators such as cylinders for providing linear motion or hydraulic motors for rotary motion). Furthermore, in addition to unmodelled effects, uncertainties due to parameter variations (e.g. uncertain fluid compressibility, load variations, changes in flow and pressure gains as well as in friction constants, etc.) and unknown external disturbances can occur. If such uncertainties, which undoubtedly can have direct impact on the bounds of achievable performance, should be considered in the controller synthesis, then robust control techniques are primarily in demand. Thus, for exact coping with inherent system nonlinearities and for providing high control performance in spite of model uncertainties, nonlinear robust control strategies are necessary for safe operations of hydraulic systems. In this paper, a nonlinear robust control approach, which is derived using a sliding mode strategy, is presented.

In view of hydraulic control aspects, a variety of control techniques have been used by different publications to

Manuscript received February 18, 2005. D. Hisseine is with the Department of Engineering Sciences, University

of Duisburg-Essen, Lotharstrasse 1, 47057 Duisburg, Germany (phone: 0049-203-379-3416; fax: 0049-203-379-3027; e-mail: hisseine@ uni-duisburg.de).

control hydraulic actuation systems. Among such methods, one finds e.g. nonlinear control concepts based on variable structure control schemes [3], feedback linearization techniques (e.g. [4], [5]) as well as Lyapunov techniques (e.g. [6], [7], [8], [9]). Most of the hydraulic research works available in the literature have been done for actuation systems operated by symmetric double-rod cylinders and hydraulic motors and (in some cases) treated in an unrealistic simulation environment, i.e. using only computer simulations, but not using experimental implementation like in industrial use as done in this paper, where major experiments carried out on an asymmetric single-rod cylinder testbed were performed to verify the simulation results. Moreover, in the synthesis of the proposed controller, a lot of considerable parametric uncertainties such as load variations, changes in fluid compressibility (i.e. uncertain bulk modulus) and in friction constants are here explicitly considered.

As already mentioned, we present in this paper a sliding mode control approach to the position tracking for a hydraulic servosystem, which consists mainly of a servovalve and an asymmetric single-rod cylinder. Based on the state space formulation, the sliding mode control [10], also called variable structure control, is a nonlinear robust control method. Unlike distinguishing properties such as robustness against parametric uncertainties and external disturbances, the use of switched control encounter the drawback of chattering phenomenon, which may be a major obstacle in practical applications. Therefore, in this work, in order to overcome the undesirable control chattering due to the high frequency switching, the control action is smoothed according to the concept of boundary layer [11], which approximates the ideal relais characteristics to saturated amplifier characteristics. As pointed out in [12], to conform the sliding mode requirements and particularly to achieve perfect tracking while maintaining robust closed-loop performance, the controller synthesis procedure have to be performed on the basis of an appropriate sliding surface designed on a suitable canonical form. Thus, as starting point for derivation of the proposed controller, the considered hydraulic system is at first transformed into a suitable canonical form using differential geometric methods [13].

The effectiveness relating to the robustness against considerable parametric uncertainties and to the tracking performance of the presented nonlinear control approach is demonstrated both by simulations and experiments.

Robust Tracking Control for a Hydraulic Actuation System Dadi Hisseine

H

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MC2.3

0-7803-9354-6/05/$20.00 ©2005 IEEE 422

II. MODELING

The hydraulic actuation system under consideration is available as an experimental setup at our Institute Laboratory and depicted in Fig. 1, where the valve–cylinder configuration is extra shown schematically in Fig. 2. Therewith, the considered plant consists mostly of an asymmetric single-rod cylinder (also termed differential cylinder) and a servovalve. For the derivation of the mathematical model of this hydraulic system, some (simplifying) assumptions are made; among other effects of leakage flows and valve hysteresis are negligible.

The piston motion equations can be written as:

RBBAApg fpApAxm (1)

where BAflpg VVmm is the total mass, px the piston position, iA the piston surface area in the chamber iand ip the pressure in the chamber i (with BAi , ), Rfthe friction forces.

The fluid flows ),( vAA xpQ and ),( vBB xpQ in the cylinder chambers A and B respectively are given by the following flows equations:

TATAvV

AAvVA

PpPpxsgB

pPpPxsgBQ

sgn

sgn 00 , (2a)

TBTBvV

BBvVB

PpPpxsgB

pPpPxsgBQ

sgn

sgn 00 , (2b)

where the function vv xxsg )( for 0vx and 0)( vxsg

for 0vx , )(sgn the sign -function, vB the discharge coefficient of the valve orifices, 0P the supply pressure and

TP the tank pressure, vx the valve spool position. Neglecting fluid leakages, taking fluid compressibility into account and using the continuity principle, the pressure building in both cylinder chambers can be described by the following pressure dynamics equations:

pAApApA

AflA xpQxA

xV

pEp ,

)()(

(3a)

pABpBpB

BflB xpQxA

xV

pEp ,

)()(

, (3b)

where flE is the (isothermal) bulk modulus of the fluid (here oil) and the cylinder chamber volumes AV and BV are given by:

ApApA AxH

VxV2

)( 0 , (4a)

BpBpB AxH

VxV2

)( 0 , (4b)

where 0AV and 0BV are the initial volumes of both cylinder chambers including pipelines (or connecting lines) volumes at both sides and H is the cylinder stroke.

Note that the reciprocal of bulk modulus is termed the compressibility of fluid. The empirical value for the fluid bulk modulus can be given as [14]:

3max

2max1 logPp

EpE iflifl , BAi , , (5)

where i ( 3,2,1i ), flmE and maxP are constant. In

Fig. 1. Experimental Testbed

Fig. 2. Schematic diagram of the servovalve-cylinder configuration (without the load mounting)

423

practice, the fluid bulk modulus is often uncertain due to its sensitivity to changes in fluid temperature. Note that in this paper, among other parametric uncertainties, these parameters i are allowed to considerably vary from their nominal values (see section IV).

By modeling of the servovalve as a proportional system (i.e. neglecting valve dynamics), the mathematical model of the considered hydraulic servosystem can be given in state space form as follows [15]:

uxQBxA

xV

xEx

uxQBxAxV

xEx

xfAx

xxm

x

xx

BVp

B

fl

AVpA

fl

Rp

421

44

321

33

24

31g

2

21

~

~

)(1

(6)

where pxx1 , pxx2 , Apx3 and Bpx4 ,

BA AA is the area ratio, Ap AA and u the input voltage. Taking under consideration both the extension stroke case (for positive valve input signal) and the retraction stroke case (for negative input), the expressions

AQ~ and BQ

~ in (5) are given as:

0ufor

0ufor~

3

303

T

APx

xPxQ , (7a)

0ufor

0ufor~

40

44

xP

PxxQ

TB , (7b)

where both conditions 03 PxPT and 04 PxPT hold. The Friction force Rf consists of viscous friction vf ,Coulomb friction cf and static friction hf . Therewith, this friction force can be described in form of a Stribeck curve as follows:

h

p

C

x

hcppVpR eFFxxFxf )(sgn)( (8)

The discontinuous nonlinearities in the friction force (8) will be now smoothly approximated using the following nonlinear smooth approximation:

)(arctan2)(sgn xx , (9)

where the tuning parameter describes the approximation

quality. Then, note that )(arctan2)(sgn xxxxx .

III. NONLINEAR ROBUST CONTROL FOR THEHYDRAULIC SERVOSYSTEM

The proposed nonlinear robust controller is derived using an appropriate sliding mode control approach.

A. Preliminary Remarks In this subsection, some basic concepts of sliding mode

control will be briefly reviewed. The sliding mode control forces the system trajectories to reach and stay on a prescribed sliding surface. Let a dynamical system be given in the form

tduxgxfx , (10)

where nx , mu and with the assumption: bounded modeling uncertainties in xf , xg and unknown (but bounded) disturbances td .

The sliding surface is given and denoted by

xss , (11)

where Tmss1s is a m -dimensional switching

manifold. Then, the sliding mode controller can have the following

general structure:

xuxuxu req , (12)

where

xu eq is the so-called equivalent control, which stabilizes the nominal system and

xu r the discontinuous robust control component, which fulfils the sliding conditions

0!

ii ss , mi ,,1 . (13)

Note that finite reaching time is guaranteed by modifying the sliding condition (13) to:

iiii sss!

, 0i , mi ,,1 . (14)

B. Sliding Mode Tracking Control for the Hydraulic Actuation System For the purpose of nonlinear analysis of some structural

424

properties related to the previous hydraulic actuation system and particularly for the design of a suitable sliding surface, the considered hydraulic system (6) need to be transformed into a canonical normal form using differential geometric methods [13]. Note that (in association with the piston position as system output) the mentioned system has a relative degree 3r strictly less than the system dimension

4n , i.e. there exists an unobservable subsystem (of dimension one), the so-called zero dynamics. After some calculations, it can be shown that the transformed hydraulic system can be presented in the following nonlinear form:

,,,

zzgzfz u

(15)

where Tyyyz (due to the fact that the relative degree 3r ) and the one-dimensional subsystem

is the aforementioned zero dynamics associated with the (piston position) output pxzy 1 .

In order to achieve perfect tracking while maintaining robust closed-loop performance, the hydraulic system under consideration should be minimum phase, i.e. the mentioned zero dynamics must be asymptotically stable. From the (stability) analysis of this unobservable subsystem of dimension one and under some assumptions, it is relatively easy to find out that the minimum phase property of the considered hydraulic system is ensured (i.e. the zero dynamics is stable, namely for any equilibrium condition).

Given the desired motion trajectory, the control objective is to synthesize a control input u such that the system output y tracks the desired trajectory dy as closely as possible in spite of various model uncertainties. The sliding mode control design procedure starts from the definition of a suitable sliding surface. For the purpose of keeping stability conditions and enhancing closed-loop system performance, the following sliding surface is defined:

t

edtCeCeCes0

012 , (16)

where 0C , 1C and 2C are design parameter for the sliding surface and dyye denote the tracking error.

From the condition 0s , we obtain at first

eCeCeCyy d 012)3()3( (17)

and then the equivalent control:

x

xxx ˆ

)(ˆ)(

3

1

)1(1

)3(

i

iid

eq

eCyu (18)

with

txf

mxV

xE

xV

xE

mxA R

gB

fl

A

fl

g

K 2

1

4

1

322 ˆ

ˆ1ˆ1ˆ

ˆ)(ˆ x

, (19)

41

43

1

3 ~ˆ~ˆ

ˆ

ˆ)(ˆ xQ

xV

xExQ

xV

xE

mB

BB

flA

A

fl

g

vx , (20)

where the expressions with the superscript ˆ describe expressions with nominal parameters.

To demonstrate the robustness of the proposed controller, a lot of considerable parameter variations (especially uncertain bulk modulus, load variations, changes in friction constants) are allowed to vary from their nominal values, so that the following expressions are to be specified:

iiflmiifl Px

ExE 3max

3213, log (21)

22

222,

arctan2

exp

arctan2

xCx

FF

xxFxf

ihihic

iviR (22a)

t

xf

mxV

xE

xV

xE

mxA iR

igB

ifl

A

ifl

ig

Ki

2,

1

4,

1

3,22 11)(x

(22b)

41

4,3

1

3, ~~)( xQxV

xExQ

xV

xE

mB

BB

iflA

A

ifl

ig

ivi x (22c)

)()( 11 xVxVmm BAflipig , (22d)

where the subscript i can be min or max according to the parameter variations around their nominal values characterized through the superscript ˆ .

Starting from the reaching terms [12] and using the sliding conditions

sss , (23)

where the design parameter is strictly positive,the robust component ru of the sliding mode control law can be after some calculations derived and given as:

))((sgn)()( xxx sur , (24)

where the robust control gain x can be expressed as:

425

bb

i

iid

b eCy ˆ)(ˆˆ

1)(

3

1

)1(1

)3( xxx , (25)

where b the additive error estimation bounds given as

bxx ˆ and b the multiplicative error bounds

given as max1

minb .In order to counteract the inherent chattering phenomenon

or to obtain smooth control action, a boundary layer [11] is introduced around the sliding surface and from this measure, the robust control component (24) can be modified to

)(sat)()( xxx sur , (26)

where represents the width of the boundary layer. The derived sliding mode control law is now given by

both contributions (18) and (26) as follows:

)(sat)()()( xxxx suu eq . (27)

Note that stability proof can be ascertained by using the

Lyapunov function 2

21

sV . However, for guaranteeing

that the system trajectories have to reach the prescribed sliding surface in finite time and stay on this, the sliding conditions 0ssV have to be modified to sss as aforementioned.

To demonstrate the effectiveness of the derived control law, experiments were carried out on the experimental testbed shown in Fig. 1 as illustrated in the following section.

IV. SIMULATION AND EXPERIMENTAL RESULTSThe nominal physical parameter values of the hydraulic

servosystem under consideration are: mm63kd , 2mm23117.Ap , 04.2 ,

mm500H , kg2.604ˆ gm , Ns/m5000ˆvF ,

N50ˆcF , N700ˆ

hF , m/s0175.0hC ,3

0 cm6.198AV , 30 cm8.297BV ,

3kg/m870fl , MPa80p , MPa0Tp ,MPa1800flmE , 5.01̂ , 90ˆ2 , 3ˆ3 ,

MPa280maxP , l/min150NQ , MPa7NP ,

Nv

Nv

Px

QB

5.0max

.

The bounds of uncertainties ranges are given by the following expressions, where minmax,i :

%201ˆ gig mm , %101ˆ jij ( 3,2,1j ),

)(,, ijiflifl EE , %FF vv 101ˆ ,

%FF cc 101ˆ and %FF hh 101ˆ .

These considerable parameter variations serve to demonstrate the robustness of the proposed controller.

To determine the sliding surface design parameters 0C ,

1C and 3C , while maintaining stabilizing requirements, stable eigenvalues must be suitably selected, namely using pole placement. Thus, these parameters are determined here via assignment of all poles by -80. The width of the boundary layer is selected as 05.0 and the design parameter 1 .

To experimentally illustrate the effectiveness of the proposed nonlinear controller, major experiments carried out on the experimental setup in Fig. 1 were performed to verify the excellent simulation results shown in Fig. 3 and Fig. 4, namely under different operating conditions. For lack of place, all the experimental results can not be unfortunately presented, but only a few as shown in Fig. 5 (piston position tracking) and in Fig. 6 (pressures behavior in both cylinder chambers). These experimental results demonstrate that the presented control approach is highly effective, i.e. this control law not only provides improved tracking performance, it also ensures high robustness against considerable parametric uncertainties (especially load variations, changes in fluid compressibility and friction constants). It can be experimentally demonstrated, that the proposed nonlinear controller can achieve a much better performance as conventional PID controllers; however, the belonging experimental results are not explicitly shown here for lack of place.

0 1 2 3 4 5 6 7 8 9 10-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

time t [s]

Pisto

n po

sitio

n [m

]

Piston position: Simulation results

Simulated valuesDesired values

Fig. 3. Sinusoidal piston position tracking (simulations results)

426

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

6

7

8x 10

6 Pression pA and pB : Simulation results

time t [s]

Pres

sion

[pa]

pApB

Fig. 4. Pressures in both cylinder chambers (simulation results)

0 1 2 3 4 5 6 7 8 9 10

-0.1

-0.05

0

0.05

0.1

Piston position: Experimental results

Pisto

n po

sitio

n [m

]

time t [s]

Measured ValuesDesired values

Fig. 5. Sinusoidal piston position tracking (experimental results)

Fig. 6. Pressures in both cylinder chambers (experimental results)

V. CONCLUSION

In this paper, a nonlinear robust control approach to the position tracking of a hydraulic actuation system is proposed. This nonlinear tracking controller is derived. using a suitable sliding mode approach. The improved robustness against considerable parametric uncertainties (e.g. load variations, changes in fluid compressibility and in friction constants) and the tracking performance of the presented control methods are shown both by simulations and experiments.

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(translated from German). Aachen: RWTH Aachen, Institut für Hydraulische und Pneumatische Antriebe und Steuerungen, Dissertation, 1977.

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0 1 2 3 4 5 6 7 8 9 10-20

0

20

40

60

80

100

120Pression pA and pB : Experimental results

time t [s]

Pres

sion

[bar

]

Pression pAPression pB

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