Different Nonlinear Controllers for Hydraulic Synchronizing Cylinders

Markus Lemmen and Markus Brocker

Department of Measurement and Control, FB7/FG8, University of Duisburg, 47048 Duisburg, Germany.AND: System and Control Group, Faculty of Mechanical Engineering, Technical University of

Eindhoven, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.e-mail : lemmen@uni-duisburg.de

Keywords: Hydraulic Cylinder, Nonlinear Control, Flat-ness Based Control, Exact Linearization

AbstractHydraulic cylinders are found in many applications. To im-prove classical linear control approaches for these nonlinearplants two different nonlinear controller design methods arepresented and compared: the exact linearization techniqueand flatness based control. Both control methods show goodtracking behaviour with experimental results.

Figure 1: Hydraulic driven synchronizing cylinder

1 Introduction

Hydraulic cylinders are wide spread used as actuators formoving heavy loads. They offer a good power-to-weight-ratio and enable the designer to separate the hydraulic pumpand drive unit. They are built in a compact fashion and needlittle space w.r.t its power. In contrast to these nice con-structional properties, controlling hydraulic drives becomestedeous due to their inherent nonlinear dynamics. In prac-tice, linear controllers are frequently used for these nonlinearplants. To increase control performance nonlinear controllerdesign may be used. Two different methods are presentedand compared in this paper: the exact linearization techniqueand flatness based control.

PSfrag replacements

Figure 2: Layout of a synchronizing cylinder

2 Modelling

Hydraulic cylinders may be built as differential (also knownas single-rod cylinders) or synchronizing cylinders (alsoknown as double-rod cylinders; c.f.. fig. 1). As depicted infig. 2, synchronizing cylinders are characterized by the equal-ity of the pressure area in chamber A and B, namely , acharacteristic which does not hold for differential (or single-rod) cylinders.

The nonlinear dynamics of the hydraulic cylinder may beapproximated as an affine input state-space model (c.f. e. g.[2, 17, 8, 20]). By defining states as piston rod position

, piston rod velocity and differencepressure we obtain a case dependend

three dimensional model

(1)

where

for and

elsewhere. The friction force is computed as

sgn

Its derivative may be approximated by

3 Controller Design

The dynamic model of the synchronizing cylinder presentedin the previous section is a nonlinear one. Thus, a controllerdesigned for tracking purposes should also be nonlinear inorder to consider the plant’s dynamics correctly. Two differ-ent approaches may be used for this purpose: flatness basedcontrol, which was developed from the field of differentialalgebra (c.f.[4, 15, 10]) and exact linearization techniquesfrom the field of differential geometry (e. g. [11, 6]). Themodel (1) is single-input-single-output and flat (this state-ment will be proven later on) and thus may be controlled bya flatness based controller. However, the system also may becontrolled by state linearization (c.f. [19]). Hence, we willshow how these two approaches may be used for trackinghydraulic synchronizing cylinders. For performance evalua-tion purposes we make use of the numeric approximations ofthe following integral performance functions

and

where ; the index means reference trajectory.

a)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.58

0.56

0.54

0.52

0.50

0.48

0.46

0.44

ym

ts

yyrt

b)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.55

0.50

0.45

0.40

ym

ts

yyrt

Figure 4: Desired and measured cylinder position of the flat-ness based control: (a) correct (b) incorrect initialposition ( m)

3.1 Flatness Based Controller Design

First, we show that the system model (1) is flat. Prov-ing flatness means finding an output which is a differ-ential transcendence basis of the system model (see e. g.[5, 12, 14, 20]). Within these calculations the expression

i.e. the input expressed as a functionof the differential transcendence basis is called the flatnessbased controller. Considering the piston position as output

, we observe that the first two states immediatelyprovide flatness, because they depend on only: and

. For the last state (i.e. the difference pres-sure) we computei.e. if absence of disturbance ( ) is assumed it is pos-sible to evaluate and furthermore

without integration. Thus, is indeed atranscendence basis and thus the system model is flat.

According to the case distinction ( ) we obtain from

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plant

flatness part of the controller

flat pre-controller

offlinecalculations

Figure 3: Flatness-based controller scheme

the previous calculations the flatness based controller

(2)

otherwise. In fact, the previously introduced control law onlyensures being close to the desired trajectory. To ensure thetrajectory is followed more exactly, the trajectory error hasto be measured and the control law has to be extendedas shown in fig. 3: An additional accelerating and stabiliz-ing outer control circuit will be used. Good results can beachieved even with a simple position error feedback

(3)

The experimental results of this flatness based control ap-proach is depicted in fig. 4(a). The performance is

ms and m s. If the initial po-sition values have been estimated incorrectly the behaviour isstill satisfactory (c.f. fig. 4(b)). The desired trajectory is metwithin seconds without any overshoot.

3.2 Exact State Linearization

Finally, the plant is controlled by a controller based on ex-act state-linearization (c.f. [11, 6, 7, 18]). The controllerscheme is shown in fig. 5: The aim is to design a (nonlinear)feedback controller (and a coordinate transformation) which

transforms the original nonlinear plant into a linear and con-trollable (closed loop) system. Finally, a linear controller forthis linear system has to be designed for tracking purposes.

Since the system model (1) has relative degree

we can compute such a state linearizing controller (see [11,6]) as

L L

L L(4)

where

is the linear controller used to assign the fedback system ap-pealing eigenvalues. Thus, for (1) the controller (4) becomes

(5)

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plantlinearcontroller

linearizingcontroller

exact state linearized

Figure 5: Exact (input-state-) linearizing controller scheme

for and

everywhere else.The eigenvalues have been chosen to be

and hence, the coefficients in (5) read

The experimental results shown in fig. 6(a) prove our ex-act linearization approach (5) with a linear output feedback

to be suitable for solving the tracking con-trol problem: ms and m s.The behaviour is satisfactory even if the initial position hasbeen estimated incorrectly (c.f. fig. 6(b)). The controlled dif-ferential cylinder reaches the desired trajectory within

seconds and reasonable overshoot.

4 Comparison and Conclusions

In this paper we show that the flatness-based approach aswell as the exact linearization approach is suitable for track-ing control of hydraulic synchronizing cylinders. Withinour experiments both controllers were designed such that the

a)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.58

0.56

0.54

0.52

0.50

0.48

0.46

0.44

ym

ts

yyrt

b)

0 0.5 1 1.5 2 2.5 3 3.5 4

0.55

0.50

0.45

0.40

ym

ts

yyrt

Figure 6: Desired and measured cylinder position of the ex-act linearizing control: (a) correct (b) incorrect ini-tial position ( m)

desired sine-trajectory is met within almost similar time ofapproximately 0.57 seconds. Yet, the flatness based con-troller shows slightly better performance than the one based

on exact linearization. However, the flatness-based con-troller has to be found manually while the computation ofthe linearizing (nonlinear feedback) control may be sup-ported/automized by toolboxes based on computer algebraprogrammes (c.f. [3, 9, 16, 13] and [1]). In contrast to lin-earization techniques the knowledge of the desired trajec-tory is essential for flat control. Thus, flatness based controlseems to perform better because it exploits the knowledge ofhigher derivatives of the reference trajectory while the lin-earizing controller in our framework does not.

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