Design of Hydraulic Force Control Systems with State Estimate Feedback

Copyright © IFAC 10th Triennial World Congress. Munich. FRG, 1987

DESIGN OF HYDRAULIC FORCE CONTROL SYSTEMS WITH STATE ESTIMATE FEEDBACK F. Conrad and C. J. D. Jensen COlllml Ellgilll'erillg IlIslill/it'. Till' TI,(/lIIirai ['lIil'f)s il\'

LJllgby,

of'

[)I'II 111(11 Ii.

DI\.-2800

[)nllllarli

Abstract. The control problem to keep a desired force acting on a machine part can often be solved by using a servo controlled oil hydraulic actuator unit. This paper descr.ibes the results of an analytical and experimental investigation of some new design principles of force control systems which are based on the application of an electro-hydraulic servovalve controlling the volume rate flow to a hydraulic motor. It is shown how the system performance can be improved compared to a conventional force feedback by using the following control principles: (1) A controller with output feedback and velocity feedforward, (2) A Luenberger observer and state estimate feedback, (3) A Luenberger observer, state estimate feedback, and unity output feedback, and (4) Velocity feedforward combined with (3).

Keywords.

Hydraulic systems; hydraulic control; force control; state space methods;

observers; output feedback.

INTRODUCTION

force acting on a piston rod of a hydraulic cylinder, when the rod is disturbed by changes of the piston rod position.

Position, velocity and acceleration of heavy loaded machine parts are often controlled by applying oilhydraulic servomechanisms. The hydraulic actuators have especially many advantages in application to mobile equipment. An other important application of hydraulic control is the control of forces and torques acting on machine parts.

The chosen hydraulic actuator system consists of an oilhydraulic cylinder with a piston which can be moved bidirectionally by an electrohydraulic servovalve. A schematic diagram is shown in fig. 1. The servovalve is controlled by a current amplifier with input voltage u(t) in order to eliminate the

A typical example is when the control tasks are to control forces and torques acting on brakes. An other example is the demand to control the forces which act between a trawl and fishing boats. Signi-

time constant of the torque motor. The servovalve

controls the volume rate flow q to the cylinder chambers. The piston rod seals are designed to have a very low degree of stiction.

ficant dynamic variations in the forces have a ne-

gative influence on the catch. A third example is the force acting on a scanning wheel designed for measuring the high of each sugarbeet relative to the surface of the earth. The scanning wheel has to be in contact with the surface and the top of the sugarbeets by a constant controlled force. The position signal from the scanning wheel controls the reference input to a servo controlled cutter system of a developed prototype sugarbeet top cutting machine. In order to obtain better static-dynamic behaviour some new design principles of force control systems using a hydraulic actuator controlled by a servo valve have been investigated based on state space modelling, computer simulating and experiments.

LOAD

The performance is improved compared to a conven-

tional force feedback by using the following control strategies: (1) A controller with output feedback and velocity feedforward, (2) A Luenberger observer and state estimate feedback, and (3) A Luenberger observer, state estimate feedback and unity output feedback, and (4) Velocity feedforward combined with (3).

Fig. 1.

The main variables are the flow rate q and the hydraulic pressure difference Pm = P1-P 2'

THE CONTROL PROBLEM AND THE HYDRAULIC ACTUATOR SYSTEM

In general hydraulic systems are non-linear due to

The control problem is here to keep a constant ACl---K

Schematic diagram of the hydraulic actuator system.

30i

F. Comae! and C. J. D . .l e nse n

;)08

the non-linear pressure d rop versus flow characteristics o f va l ves and orifices and n on-linear fric tion conditions causing by stiction between spoo l s and cy li nder barrels , sea l s and p i ston rods etc . A simple approximated linear model o f the h ydrau lic actuator system is shown i n fi g. 2 . The mod e l and the constants have been identified based on measurements carried out using a static and dynamic volume rate fl owmeter FM 4000 , whi ch has been developed at the Control Eng ine ering Institute , The Technical University o f Denmark. The linear model is a third order model with three time constants. Furthermore , a non- linear mode l has been identified and used in the evaluation of the investigated control strategies . The observer and controller design has been carried out based on the given linear mathemat ical model of the actuator system , where the constants are: a

effec tive piston area

flow pressure ga i n of the s ervova lve flow gain of the servova l ve

time constant o f the servovalve

volume flow c urrent

u

con trol input voltage

v

piston rod pos ition

v

p iston rod velocity

v

piston rod accelerat i on

The system can be described by three state vari ables. It i s evident t o chose the ma in var iable fl ow and pressure difference as two of the state variables. As the t hird state variab l e we can of course se l ect dq(t)/dt. But this will give a very high gain factor in the inner fe edback loop of q(t) to q(t) . Thus we have selected the f o llowing state variables q(t) + T q(t) 2 q(t)

(6)

p (t) m

(8)

(7)

[0

° _a · ~ (t) )T

(9) ( 10)

(11 )

The fo l lowing constants are defined l/T , Cl = I/T2 2 I I/T3 = 4!3c t /V t ('(3 4 (3/ V ('(4 t which implies that CL > O, CL > 0 , CL > 0 , and

and the variables are

i

(5)

F(t)

y(t) = F(t)

resistance of coils in the torque motor

p ressure difference

(4)

(t)

v (t) -m · ~ ( t) 2 and the outpu t i s

time constant of the ser vovalve

Pm

m

VI (t)

total p ressure volume

q

p

The disturbance input var i ables are defined as

mass o f p isto n and rods

controlled force

(3)

x (t) 2 x3 (t)

le akage coefficient

F

The cons idered actua t or system is described by the the equat i o ns

xI (t)

CI+k qp

m

and the cons tant matr ices are A: (n x n) , B: (n x I), ( I x n). v (t) is disturbance on the out put. 2

c:

CL

l

Cl > 0. 4 l 3 The matrices of the state space model then become

A

( 12)

B

( 13)

c

°

= [ 0

°

c 3) = [0

( 14)

a)

For a selected hydraulic actuator unit the following values have been est imated based on measure ments with respect to a chosen working point at system pressure level at 70 bars: a 2.13 . 10- 4 m2

B

7 "1 0

8

N/m2 2 12 3 (m /s)/(N/m ) -5 3 2 1/2 (ps - Pm) · 2 . 79 ·1 0 (m /s)/(A(N/m) )

3 . 5 · 10-

m

Fig . 2 . Approximated linear model of the hydraulic actuator system.

R

Tl T2 A STATE SPACE MODEL OF THE ACTUATOn SYSTEM

1. 3 kg 470Q 5 . 7 ms 1 .2 ms

and T3

The linear system equat ions (3) - (5) can be express ed in a time invariant , single input - sing le ou tput state space model

Q

11 . 2 ms 2 (B AB A B)

- CL

b x(t) y(t)

Ax(t)+BU(t)+V (t) I cx(t)+v (t) 2

(I) (2)

l

l: ° CL

l

2

CL - CL

l 2

'

(Cl

l

+CL ) 2

1

(15)

CL Cl

2 4

which implies that the system is control labl e for bl'''O. This is always obtained because k q >0 f o r

Hydraulic Force Control Systems Ps>Pm' where Ps is the supply pressure. The observability matrix is

P

( 16)

\-/hen r (t) = 0 we have a regulator design problem of a single Input - single output system. Thus we can use the Ackermann formulas (1972) to compute the control gain matrix L and the observer gain matrix K, i.e. L

which implies that the system always is observable because c = a .. O. 3 DESIGN PRINCIPLES OF FORCE CONTROL SYSTEMS

The most simple force control system is obtainedby a conventional output feedback, i.e. measuring the force F with a transducer and close the loop using a P-controller or PI-controller. In this case the design can be based on the equation 2 (K/kt)r(s)-[ (1+T S) (1+T s)A /ctlsv(s) 2 1 F(s)= (17) where the gain K=Aktkq/(RC ) and k is the forcet t transducer constant (or gain). The reference input is r(s) and sv(s) is the velocity disturbance. Here, the acceleration disturbance input has been neglected because ~(t) is the most domineering on the piston rod when v(t)=vOsinwt. In general only a poor performance can be obtained by such a control system. Force Feedback and Velocity Feedforward A much better design solution to the force control problem is obtained by using both a force feedback and velocity feedforward. The feed forward transfer function can approximately be described by

v

(s)

Ra K

=

e T .Q-1 S (A) n cv

(21 )

where e~ = [0 0 ... 0 lity matrix and S (A) = An_S An - 1 _ cv n-l

T

ll , Q is the controllabi(22)

The coefficients Si are given by the chosen poles of the controller. Furthermore

FOrCe Feedback

G

309

(18)

q

(23) where P is the observability matrix and n n-1 YOv(A) = A -Y n _ 1A

... -YjA-YOI

(24)

The coefficients Y are given by the chosen poles i of the observer. Defining x(t) = x(t)-x(t), it follows that the regulator design problem can be solved by solving the equation :(t)] [

x(t)

[A-BL

BL A-KC

0

]

[~(t) 1

(25)

X(t)J

which describes the behaviour of the closed loop system. The characteristic-equation roots of this system consist of the sum of the observer poles and the controller poles (the separation principle) . For the actual hydraulic actuator we have L (1 12 1 ) and K = (k k2 k3)T. 1 1 3

Observer, State Estimate Feedback and unity Output Feedback

Observer and State Estimate Feedback By using the full order Luenberger Observer (1964, 1966) new principles have been developed to the design of force control systems based on a hydraulic actuator controlled by an electro-hydraulic servovalve.

Further the static performance can be improved significantly by introducing a co-state during an output unity feedback loop (Davison and Wang, 1975; Hansen, 1983; Conrad, Hansen and Trostmann, 1984) and an additional integrator as illustrated in fig. 4.

The estimated state vector x(t) is fed back. Thus the state space model of the closed loop system is given byeqs. (1) and (2) combined with the observer equation and the control law equation x(t)

AX(t) + Bu(t) + K(y(t)-Cx(t))

(19)

u(t)

-Lx (t) + Hr (t)

(20)

Fig. 4.

State estimate feedback, unity output feedback and velocity feedforward.

In the design we demand that if v(t)=O then lim[r(t)-y(t) t+=

Fig. 3.

Force control with state estimate feedback.

1

= 0

when

lim r(t) = rO t+=

where r(t) is the reference input, here a scalar, and rO is a constant.

310

F. COll ra d all d

The actuator system equatio ns (1) and (2) and the observer equation ( 19) must be combined with the f o l l ow ing equations r(t) - y(t)

(26)

- Lx(t) + 14z(t)

(27)

where z(t) is the co- state, here a scalar , and 14 an additional feedback gain factor, here also a scalar . To find the controller gain matrix and t he observer gain matr ix we let v(t)=O . Then eqs . ( 1 ) , (2) , (20) , and (2 1) give

x'

A'

(t)

B'

w

[ -L-1 ]

4

[X(t) ] =[-v x']

D . .I CIlSC Il

ferent design principles . A range of s i mulations and experime nts has b een carried ou t . Some of the results are discussed h ere and some illustrative responses obtaine d with the designed contro ll ers a nd observers applied together with the non - linear model and the rea lized f o r ce control sys tems. Control Sys tem with Force Feedback As expected we obta in poor perfo r mance wi th onl y force feedback , which appears from both the s i mulated stepresponses in fig. 5 . and the measured respons es in fig. 6 , and furthermore f rom the d i s turbance responses in fig . 7 . The disturbance input is v(t) = vOsin2 TIVt with V = 1 Hz and vo = 0 . 03 m. Further it appears that there is a good agreement between the simulation and meas ur e d results .

and

u(t)

c:. J

(29)

z (t ) Def i ning x(t)=x( t )-x(t) and combining these equa tions with the observer equation (25) we find the stat e space model of the closed l oop servosystem as

: ' (t) [

I = [ A'- B ' L '

x(t)

y(t)

0

= [c

0

B'L'][~ ' (t) ] A- KC

x(t)

+

[W ] r (t)

(30)

N

;-,

600

,",

/

~oo

I

I

force feedback

', /

-----------

'

\

I I I

400

0

1 [ X(t) ] J z (t)

,UU

slale e,li male feedback

300 (31 )

The state vector x(t) and the constant matrices A, B, and C a r e the same as in sect i on 3 . From eqs . (32) and (33) it appears that th e considered servo d es i gn probl em can be tra nsformed to a r egulator design prob lem for a single input single output system. Thus applying the Ackermann formu l as , L ' and K can be computed , when t he contro l ler po l es and observer poles h ave been chosen.

200 100

o

o

/

S IMULATION AND EXPERIMENTAL RESU LTS

0.0 1

0.02

0.03

0.04

0.05

Time

Fig . 5 .

Simulated stepresponses f o r the control system with respectively f orce feedback and state es timate feed back .

Observer, State Estimate Feedback , unity Output Feedback and Velocity Feedforward The performance can furthermore be improved signi ficantly by adding a velocity feed forward l oop to the contro l system with state estimate feedback and unity ou tput feedback. The feed f orward gain is Ra/Kq , because even if the characteristic of the control system has been changed with the state estimate feedback , the disturbance is st ill working on the unchanged phys i cal actuator system .

sec

,00

N

fn rce fl>eclback

600 ~no

400

:lOO

The mathematical models presented in the previous sectio ns have been programmed f or compute r simu l a tions in order t o study the obtained performance with the di ff erent control s t rategies. Furthermore the designed controllers and observe rs have been combined wi th a non - linear model of the hydr aulic actuator system. The main non - linearities are the fl ow equat io ns and the fricti o n forces .

200 100

sec Cl

0

0.0 1

0.02

0.03

0.04

0.05

Time:'

The models inc lude only v is cous friction . In the experimental set up the stiction has been elimi nated by using special designed seals of f luorcar bon o Another non-linearity is the time delay , caused by the fluid channels between the servovalveand the cylinde r. In the actual case the time delay is estimated to only about 0 .3 ms which will not have a significant influence on the dynamic behaviour .

Control System with Force Feedback and Velocity Feedforward

In the laboratory an expe rimental set up has been built in order to evaluate the mode l s and the dif -

The performance can be significantly improved by adding the ve l ocity feed forward loop. This appears

Fig . 6 .

Measured stepresponses for the control system wi th respectively force feedback and state estimate feedback.

H H lra uli c Fo rce CO ll lro l Syste m s

3 11

from the dis t u r bance responses shown i n fi g . 7 . The amplitude of the simulated and the measured response is 5 % of the set point f o rce on 500 N.

The cons i dered control system has the d i sadvant age that even smal l amplitude of disturbances will re sult in osc i l l ati ons of t he force output.

By addi ng t he veloc i ty feedforward loop , i t is not poss i ble to compensate for the time cons t ants of the servovalve . This implies that distu r bances at high frequenc i es, will not lead to a con tro l system wi th a sat i sfactory performance .

Contr ol Syst em with State Estimate Feedb a c k and Un i ty Ou t p ut Feedback

;i

1100

Applyi ng this contr o l str ategy , the performance can be impr oved very s i gn ificant which clear l y appea r s f rom t he s i mu l ated responses shown in fi g s. 9 a nd 10 . The s e tt l i ng t i me is about 15 ms with state estimate f eedback and increases to about 20 mswhe n we are add i ng a unity output feedback loop. However this wil l improve the performance since the force a mp l itude , caused by the di s turbance i s r educed from 110% t o 2% of the set point va lue .

...... without velocity feed f o rward :i. 7oo w

~IOO

.- . .............. ..... ........................... . .. , .•.•............. . ..

~

l

\

soo

with vel oc ity feed forward

zo.

400 -I--~--r-----'--- 1 ----- ' -'-' 1 -' 0..

0.'

- .-- - , - - - - -, 0. '

withou t u n ity feedback

! with unity ou tput feedback

.00

1.0

TIME :SEC .

%00

Fig . 7 .

100

Simulated forceresponses for position input at 1 Hz for the control system with force feedback , r espectively with and withou t velocity f eedforward.

0.00

0.01

0. 01

O.OJ

0.04

0.05

TIME SEC.

Control Sys t em with State Estimate Feedback The poor perfonnance of the force feedback system can also be improved by adding the third order Luenberger obser ver d es cribed in section 4 . The obtained stepresponse with o bserver and state esti mate feedback is shown in fig. 8. The force amp l i tude due to the disturbance is 130 % of the set point force on 500 N without observer (see fig. 7) and o nly 11 0% with the o b serve r and state estimate feedback as it appears from fig . 8 .

Fig. 9 .

S i mulated stepresponses for the contro l system with state es timate feedback re spectively wi th and without un i ty output feedback .

z·oo w

:i o ~

Z

..o

unity output feedback

1'[00

"0

\

/ ~ ..... .. .-~ .. ...... .. .. .... /·state es timate feedback force feedback and velocity f eedforward

v elocity feed forward

400-l--~--,--~--,---_--r--_--r------, 0.1 0.0 0.4 0.% ' .0 0.'

TIME SEC •

•

-I--~--·r-0.0

0-'-

• . - .- -,--- - .. . -- T- ·--· ~-r ---,---.., 0. .

0 .'

0. '

1.0

TIME SE C.

Fig . 8.

Simulated forcerespons es for position input at 1 Hz , f o r the control system with respectively f orce feedback with ve l ocity feed forward and with state estimate feedback .

All the three controller po l e s are placed in - 500. In the experiments with the actual system i t has been shown that the poles only can be moved to a limit a t -800. This means that the t i me constants shou l d be at least 3 - 4 times the delay t i me.

Fig. 10 .

Simulated forceresponses for position input at 1 Hz , for the control system wi th r espectively force feedback wi t h velocity feedforward and state estimate feedback with unity output feedback .

Control System with State Est imate Feedback , Unity Output Feedback and Velocity Feed forward Figure 11 shows h o w much the disturbance inf l uence wil l be reduced by adding a velocity feed f o r ward loop . The amplitude is r e d uced to 2% of the set point va l ue , which mea n s that even a large disturbance o f low f requency has a nearly i ns i g n i f icant impact on the output .

F. Conrad and C.

312

tl

Hansen, N.E. (1982). On Stochastic Control and Optimal Parameter Estimation of the Bending Process. Ph. D.-thesis. Technical University of Denmark, Control Engineering Institute.

without velocity feed forward

...'" 510

...

........ . .......... .................... . .

Jensen, C . J.D. (1984). Oilhydraulic force regulators with observer (in Danish) master thesis report, Technical University of Denmark, Control Engineering Institute, S84.29. Luenberger, D.G. (1964). Observing the state of a Linear System. IEEE Trans Mil Electron, Mil-8 , 74-80.

'ID

with velocity feed forward

~+----r---,----~--,----r---,----~--,----r---. 0.0 0.% 0.4 0.1 '.0

0.'

TIME SEC.

Fig. 11.

D. Jcnsen

Davison, E.J., S.H. Wang (1975). On pole assignin linear multivariable systems using output feedback, IEEE Trans. Aut~ Control, Vol . AC-20, pp. 516-518.

Zlro

o

J.

Simulated forceresponses for position input at 1 Hz, for the control system with state estimate feedback and unity output feedback, respectively with and without velocity feedforward.

This small deviation between the reference input and the output, has to be seen relatively to that the amplitude of the disturbance is about 5 times larger than the leakage flow in the hydraulic actuator. CONCLUSION The presented results of the investigation of the new design principles of force control system applying a hydraulic actuator show how the performance can be improved significant by the control strategy based on state estimate feedback and output unity feedback. The best solution is obtained when this is combined with a velocity feedforward loop. It appears that the developed mathematical state space models provide a powerful tool for design of an actual force control system. ACKNOWLEDGEMENT The results present in this paper belong to a development project which is based on the results from a master thesis project carried out by Carl J .D. Jensen (1984 ) at the Control Engineering Institute, the Technical University of Denmark with associate professor F. Conrad as supervisor. The two authors want to thank professor E. Trostmann, Ph.D. for his assistance during the application of the control strategies. We also want to thank technical assistants P. Lundgreen and A. S~lby for their assistance

REFERENCES Ackermann, J. (1972). Der Entwurf Linearer Regelungssysteme im Zustandsraum. Regelungstechnik und Proze8datenverarbeitung, 7, 297-300. Christensen, G.K., F. Conrad, E. Trostmann (1986). Design of Hydraulic Servoes with Observer and Unity Output Feedback, Paper 23, 7th Int. Fluid Power Symposium, Bath, organized by BHRA. Conrad, F., N.E. Hansen, and E. Trostmann (1984). Lecture Notes. Modern Control Theory(in Danish) course no. 8203. Technical University of Denmark, Control Engineering Institute, S84.23.

Luenberger, D.G. (1986). Observers for Multivariable Systems. IEEE Trans. Auto. Control, AC-11, 190-7.