Nonlinear Modeling and Control of Hydraulic Actuators

8/10/2019 Nonlinear Modeling and Control of Hydraulic Actuators

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after putting this formula into Eq. (3)we obtaintheequations

oilc

coil 1

oil

d

d

d

dp A

x

tV A x

p

p

p

t

p Q Q

1 1 0 1 11

1 1

,

i

nt

,

i

oil

c

c

oil 2

oil

d

d

d

dp A

x

t V A x

p

p

p

t

p Q

2 2 0 2 2

2

2 nt Q Qext 2

(4)

d

d

d

d

d

d

1 oil

c

c

2 o

p

t

p

V A xQ A

x

tQ

p

t

1

0 1 11 1

,int

il

cext

c2

d

d

p

V A xQ Q A

x

tQ

2

0 2 22

,int

(5)

where

Q C x p p p p

x p p p p

Q C

1 0 1 1

0 0

2 0

(

)(

sgn

sgn

sg

sg

s s s

s 1 1

sg

sg

s 2 2

s s s

x p p p p

x p p p p

sgn

sgn )

0 0

2 2

(6)

Q k p p

Q k p

int int

, ,

1 2

2 2 2ext ext

where coefficientskintandkext, 2can be evaluated by meansof the general formula by [12]

k

dr r

L

C Q

p

leak

ic c

3

N

N

6

2

050

.

Eoil[1] can be given as:

E p E C p

pCoil oil, max

1

2 10 1 2log

max(7)

and the following notification is used

sgn xx

x

x

1 0

0 0

1 0

sg x x xx

00 0

The forces on the cylinder can be expressed as (usingNewtons Second Law of Motion)

m xt

p A p A F v Fd

d

cF c dist

2

2 1 1 2 2 (8)

wherev x

tc

cd

d .

The friction forces F vF c are viscous friction, Coulombfriction and static friction, i.e.,

F v k v v F F v

CF c v c c c s

c

s

sgn exp

The plot of friction versus velocity is shown in Fig. 2.The factors of friction forcesFv (viscous),Fc (Coulomb) andFs

(static) can be identified from the total friction (FF) and veloc-ity (vc). These (FFandvc) can be estimated from the values ofthe position, and pressures in both chambers of the cylindersexperimentallywith zero load, andusing then numerical inte-gration of the position for determing velocity.

Spool valve

Spool valve dynamics can be derived ina similar way as fora cylinder, but the following linear second order differentialequation is a widely used and sufficient approximation:

d d d ds n s n2 n2

2

2 2x t

tB

x t

t u t (9)

Typical values of spool valve parameters are: natural fre-quency n 300500 s

1 and damping factor B 0.71.0.Correct modelling of a spool valve requires resetting of theintegrators by bringing their inputs to zero as soon as the endpositions of the spool valve are overstepped. This precautionis shown in Fig. 4.

State space form of the hydraulic actuator

Let x x x x

t

x p x p x x x x

t

1 2 3 1 4 2 5 6 c c

s sd

d

d

d

, , , , ,

then from Eq. (59) we get:

,

x x

xm

x A x A F x F

x E x

V A x

1 2

2 3 1 4 2 2

33

0 1 1

1

F Load

oil

1

2

oil

1

Q x x A x Q x x

x E x

V A x

Q

1 3 5 1 3 4

44

0 2 2

, ,

int

,

int x x Q x A x Q x xx x

x u B x

4 5 4 2 4 5

5 6

6 2

, ,

ext 2 2

n2

n 6 n2 x5

(10)

42 Czech Technical University Publishing House http://ctn.cvut.cz/ap/

Acta Polytechnica Vol. 42 No. 3/2002

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-200

-150

-100

-50

0

50

100

150

200

Velocity [m/s]

Friction Force [N]

Fig. 2: Friction verses velocity profile


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3 Modelling in MATLAB/Simulink

The model has been implemented in versatile softwareMatlab/Simulink, which is widely used in control engineer-ing communities around the world. Cylinder, spool valveand controller models are shown in Fig. 3. These blocks

were created using the above derived equations, and a GUI(graphical user interface) will appear by clicking the block.Parameters can be entered through these menus. Fig. 4 showsthe spool valve model, where anti-windup has been included.Fig. 5 shows the cylinder model, which is modeled using fourstate variables, and the cylinder stops at the ends have beenintroduced.

Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43


Fig. 3: Simulink block diagram of the hydraulic actuator model

Sign

Product

0

Constant

1/w0/w01

In1

Fig. 4: Simulink spool valve model with end position stops

Fig. 5: Simulink block diagram of the hydraulic cylinder model


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4 Fuzzy Logic Controller (FLC)Looking for an algorithm that would be closer to the time

optimal controller (bang-bang control), we tried using non--linear Fuzzy Logic Control design techniques. The basicstructure of a Fuzzy Logic Controller is shown in Fig. 6. From

the many types of fuzzy controllers, e.g., PID-like, slidingmode, Takagi-Sugeno etc., we selected Mamdani type FuzzyLogic Controller. This controller uses two inputs (controlerror and its derivate) and one output (control signal). Forboth inputs, three fuzzy sets (two Sigmoid and one Gaussianmembership functions), and for output five fuzzy sets (twoSigmoid and three Gaussian membership functions) havebeen taken, as shown in Fig. 6. Rules are defined such thatfor the region where we have large errors, a constant andmaximum allowable control signal will be delivered, whereasin thesmall error region it will behaves as a non-linear PD-likecontroller, see Table 1.

Two types of Inference Engines are used; one is composi-tion-based inference and the other is individual-rule-basedinference. In the defuzzification, the set of modified controloutput values in the form of fuzzy sets is converted intosingle point-wise values or crisp values. Various defuzzifica-tion methods can be used; we used the centre of gravity orcentre of area method.

Fuzzy controllers have the disadvantage that it is difficultto tune the large number of parameters, in contrast to theproportional derivative (PD) controller, which can be expres-sed by the equation:

u K e K ePD P D (11)whereuPD is the control signal, e the control error, andthe constants KP, KD represent proportional and derivate

constants. The controller parameters can be optimised

heuristically, but good support for optimising these para-meters is provided by the Matlab toolbox called the NCDBlock set.

Even in the presented case of a fuzzy logic controller whena minimum number of fuzzy variables and rules have beenused, better results than with a PD controller have beenachieved. As a fuzzifier, three fuzzy sets (two sigmoid and oneGaussian membership functions) have been used to convertthecrisp valuesof theinputs into fuzzy linguistic variables, seeFig. 9. Similarly five fuzzy sets (two sigmoid and three Gaussi-an membership functions) have been used to convert fuzzylinguistic variables into crisp values of the outputs as a controlsignal.



Normalization

Crisp process-state

values

Fig. 6: The structure of a Fuzzy Knowledge Based Controller

Control error

N Z P

Change ofcontrol error

N .. .. P

Z .. Z ..

P N .. ..

.. NL .. PL

Table 1: Lookup Table (explaining rules)

Fig. 7: Inserting FLC into the Simulink Model

Fig. 8: GUI for Output in Matlab for FLC

Fig. 9: Schematic diagram of fuzzy inference system


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The whole process of fuzzy control can be expressed in thefollowing steps, considering x as input and y as output foreach step. Also two types of membership functions are used asgiven below:

Sigmoid membership function:

f x a c e a x c ,

1

1

Gaussian membership function:

f x a c e

x c

a ,

2 2

Step 1:Fuzzifier: Conversion of crisp values to fuzzy variables:

For Input 1,i.e., error, conversion has been done using

three fuzzy variables (Negative (N) using the sigmoid mem-bership function of parameters [2500.02], Zero (Z) usingthe Gaussian membership function of parameters [0.015 0],and Positive (P) using the sigmoid membership function ofparameters [250 0.02]).

Thesame procedure was followed for Input 2, i.e., changeof error with parameters ([900.05], [0.04 0], [90 0.05]).

Step 2:Logical Operators

The logical operator AND used in the rules is describedas:

y x x min , ,1 2

Step 3:Implication

The implication used in rules is as follows:if error is P AND change of error is N thencontrolsignal is P.

Step 4:

Aggregation:In Aggregation, the rules directed towards one output are

combined together. We used the logical operator OR withmax function as given below:

y x x max , ,1 2In our case we had one output and five rules.

Step 5:Defuzzification:

Defuzzification is the process of converting fuzzy variablesto crisp values. The center of gravity method has been used toobtain crisp values of the output from the reference outputfuzzy sets. The reference fuzzy sets consist of five fuzzy vari-ables (negative large (NL) using the sigmoid membershipfunction [175 0.12], negative (N) using the Gaussian mem-bership function [0.008 0.05], zero (Z) using the Gaussianmembership function [0.008 0], positive (P) using the Gaus-sian membership function [0.008 0.05] and positive large(PL) using thesigmoid membership function [175 0.12]). Thefollowingformula hasbeen used to convert thefuzzy variablesto crisp values.

y

c a x

a x

ij ij i

i

p

ij i

i

p

1

1

As shown in Fig. 8, GUIs has been used in Matlab, whereall the input and output fuzzy variables, inference system,fuzzifier and defuzzifier can be defined very easily by settingthe corresponding parameters. Then the controller is con-nected to the model in Simulink, as shown in Fig. 7.

5 Results and conclusionsThis work has two aims: the first is to elaborate a way of

non-linear modelling of a differential hydraulic actuator suit-

45


Fig. 10: GUI for defining Rules in Matlab for FLC

0 0.5 1 1.50

10

20

30

40

50

60

70

80

90

100

110Position [mm] with Small Gain

Time [s]

X - Desired

X - PD

X - FLC

0 0.5 1 1.50

10

20

30

40

50

60

70

80

90

100

110Position [mm] with Large Gain

Time [s]

X - Desired

X - PD

X - FLC

Fig. 11: Position trajectory of hydraulic actuator for FLC and PD control


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able for programming in Simulink. The second goal is to usethis model for comparing a classical and a non-linear fuzzylogic control loop design. The obtained results do not showany discrepancy with our experience or with the availableexperimental data. The responses that we obtained confirmthe general advantage of FLC in quicker responses, due tothe fact that only FLC is able to produce control signalsthat are not only proportionally dependent on control error

as in the case of a PD controller. The fundamental import-ance of a proper choice of the gain in the PD controller isdemonstrated in by the columns in Fig. 13, where the leftcolumn shows responses with half of the gain used in theexperiments depicted in the right column. PD for small stepchanges surprisingly achieved worse control responses thanfor larger step changes. The anti-windup problem has beensolved, as shown in Fig. 15, which explains the importance of



0 0.5 1 1.5-50

0

50

100

150

200

250

300

350

400

450Velocity [mm/s] with Large Gain

Time [s]

V - PDV - FLC

0 0.5 1 1.5-50

0

50

100

150

200

250

300

350

400

450Velocity [mm/s] with Small Gain

Time [s]

V - PD

V - FLC

Fig. 12: Velocity trajectory of hydraulic actuator for FLC and PD control

0 0.5 1 1.5-50

0

50

100

150

200

250

300Control Signal [mm] with Large Gain

Time [s]

U - PD

U - FLC

Limit (160 mm)

0 0.5 1 1.5-50

0

50

100

150

200

250

300Control Signal [mm] with Small Gain

Time [s]

U - PD

U - FLC

Limit (160 mm)

Fig. 13: Control signal for a hydraulic actuator applying FLC and PD control

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Velocity [mm/s] with and w ithout anti-windup

Time [s]

With Anti-windup

without Anti-windupwith no Matlab limits

Fig. 14: Velocity profile with windup problems

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

10-3 Position [mm] with and without Anti-windup

Time [s]

With Anti-windupwithout Anti-windup

with no Matlab limits

Fig. 15: Position curves with windup problems


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respecting the limits in the model. However in this case lessdifference is shown between with and without antiwindup,due to the unsuitability of the parameters.

AcknowledgementThis work hasbeen supportedby GA R,Research Project

No: 102/01/1347; Numerical methods for optimal controlsystem design.

References[1] Berzen, W.:Nonlinear Control of Hydraulic Cylinders. Pro-

ceedings of the 6th IEEE Mediterranean Conference onControl and Systems (MCCS98), Italy, 1998.

[2] Blackburn, J. F., Reethof, G., Shearer J. L.:Fluid PowerControl. Massachusetts, (U.S.A.): The MIT Press, 1960.

[3] Costa Branco, P. J., Dente, J. A.: The Application ofFuzzy Logic in Automatic Modelling Electromechanical Sys-tems. Fuzzy Sets and Systems, 1998, Vol. 95, No. 3,p. 273293.

[4] Daugherty, R. L., Franzini, J. B., Finnemore, E. J.:FluidMechanics with Engineering Applications. Singapore: McGraw-Hill Book Company, 1989.

[5] Dimiter, D., Hans, H., Michael, R.: An Introduction toFuzzy Control. Berlin: Springer-Verlag, 1996, ISBN

3-540-60691-2.

[6] Horacek, P., Binder, Z.:An Approach for Design and Imple-mentation of Fuzzy Controller. In: Proceedings of Euro-pean Congress on Fuzzy and Intelligent TechnologiesEUFIT93, Germany, September 1993, Vol. 2,p. 163169, ISBN 3-86073-176-9.

[7] Jan, J. A., ulc, B.: Conventional and Fuzzy Logic Controlof Non Linear Hydraulic Actuators For Flexible Robots.IASTED International Conference Applied Simulationand Modelling (ASM 2001), Marbella, Spain, 2001,p. 302307, ISBN 0-88986-311-3.

[8] Jan, J. A., ulc, B.:Fuzzy Logic Control in Hydraulic Actua-tor Modelled from Own-built Simulink Block Library. Pro-ceedings of theConference on Information Engineeringand Process Control, CTU, Prague, Czech Republic,2001, p. 7374, ISBN 80-902131-7-0.

[9] Jan, J. A., ulc, B.: Modelling and Simulation of Non-linear Hydraulic Actuator Using Conventional and FuzzyLogic Control. 16th International Conference on Produc-tion Research (ICPR-16), Prague, Czech Republic,2001,ISBN 80-02-01438-3.

[10] Jan, J. A., ulc, B.: Fuzzy Logic Control in DifferentialHydraulic Cylinders driven by Servo-valves: Simulation andVisualization. 5th International Student Conference,CTU in Prague, Prague (Czech Republic): Poster 2001.

[11] Kugi, A., Schlacher, K., Keintzel, G.: Position Controland Active Eccentricity Compensation in Rolling Mills.Automatisierungstechnik Germany, 1999, at 8/99,ISBN 3-486-23477-3.

[12] Merrit, H. E.:Hydraulic Control Systems. New York: JohnWiley & Sons, 1967.

13] Noskievi, P.: Modelling and System Identification. (In

Czech), Ostrava (Czech Republic): Montanex a.s., 1999,ISBN 80-7225-030-2.

[14] Stadler, W.: Analytical Robotics and Mechatronics. NewYork: McGraw-Hill Inc., 1995, ISBN 0-07-0060608-0.

[15] Sugeno, M., Yasukawa, T.: A Fuzzy-Logic-Based Approachto Qualitative Modeling. IEEE Trans. on Fuzzy Systems,February 1993, Vol. 1, No. 1, p. 731.

[16] ulc, B.: Integral Wind-Up in Control and System Simula-tion. In: Control Engineering Solutions. A Practical Ap-proach (P. Albertos, R. Strietzel and N. Mort (Editors)),IEE, London 1997, p. 6176, ISBN 0-85296-829-9.

Bohumil ulcphone:+420 224 352 531fax: +420 233 336 414e-mail: [email protected]

Ing. Javed Alam Jan, Ph.D.e-mail: [email protected]

Department of Instrumentation and Control EngineeringCzech Technical University in PragueFaculty of Mechanical EngineeringTechnick 4

166 07 Prague 6, Czech Republic

Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 47


x1,xc position of cylinder piston [m]

x2,vc velocity of cylinder piston [m/s]

x3,p1 pressure of oil in chamber 1 [Pa]

x4,p2 pressure of oil in chamber 2 [Pa]

x5,xs position of valve piston [m]

x6,vs velocity of valve piston [m/s]

Q1/2 flow in pipe 1 and 2 [m3/s]

V0,1/2 volume of Oil in chamber at xc 0 [m3]

FF friction force [N]Eoil bulk modulus of oil [Pa]

A1 piston area in chamber 1 [m2]

A2 piston area in chamber 2 [m2]

d diameter of piston of cylinder [m]

rc radial clearance between piston and cylinder [m]

L length of piston head [m]

B spool valve damping [1]

n spool valve natural frequency [s

1

] dynamic viscosity [Pa s]

Table 2: Lists of Symbols

Nonlinear Modeling and Control of Hydraulic Actuators

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