Feedback Control of an Electrohydraulic Servosystem with Pressure-Dependent Uncertainties

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

FEEDBACK CONTROL OF AN ELECTROHYDRAULIC SERVOSYSTEM WITH PRESSURE-DEPENDENT UNCERTAINTIES K.C. Un* and K.N. Reid** *Engineering North 218. Olclahoma State University. Stil/water. OK 74078. USA " Engineering North J JJ. Olclahoma State University. Std/water. OK 74078. USA

Abstract. An electrohydraulic servosystem has dominant nonlinearities due to the valve orifice flow characteristic and the pressure dependence of the hydraulic fluid bulk modulus. Feedback control design is difficult because of these nonlinearities and related uncertainties in the system parameters. A method for feedback control design is presented which uses a linear model structure with signal-dependent parameters to present the nonlinear model of the open-loop servosystem. The method is implemented in the discrete-time domain and results in a digital controller with signal-dependent gains. The resulting feedback control system is robust for a wide range of operating conditions.

Key Words. Controller with signal-dependent gains; state variable feedback via observer and unity output feedback control; system identification

nonlinearities (Finney et ai, 1985). The hydraulic servosystem is such a case.

1. INTRODUCfION Electrohydraulic servosystems are often used for precision motion or velocity control when it is necessary to achieve a high speed of response, a high horsepower to weight ratio, and a high system stiffness. Such servosystems are commonly designed based on linear system techniques with the result that high speed of response and adequate degree of stability and static stiffness can only be achieved for a limited range of operating conditions.

In this study, a "linear" controller with pressuredependent gains is proposed to solve the above nonlinear compensation problem. The controller is designed based on an open-loop system model which is linear but has pressure-dependent parameters (Lin et ai, 1993). It is demonstrated in this paper that such a controller produces robust performance over a wide range of operating conditions.

One approach to reduce the operating limitation of a linear controller is the incorporation of a mechanism into the control loop for continuous adaptation to changes in the operating conditions. For example, self-tuning control (Vaughan et ai, 1986; Daley, 1987; Watton, 1990) and model reference adaptive control techniques (Sung et ai, 1987; Edge et ai, 1987) have been applied for the control of hydraulic systems which experience large changes of supply pressure, load inertia, or motor shaft loading during normal operation. However, continuous adaptation techniques may not be practical in cases where the system under control has a high speed of response and at the same time exhibits dominant

2. ELECfROHYDRAULIC SERVOSYSTEM A schematic diagram of a computer-controlled electrohydraulic servosystem is shown in Fig. 1. The system consists of an electrohydraulic servovalve, a motor, and a pump which supplies pressurized fluid to the servovalve. The system output is motor speed. The servovalve is assumed ideally critical center with matched and symmetrical orifices. An empirical equation for the valve orifice flow rate is (Merrit, 1967):

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where J, is the effective polar moment of inertia of the motor and load; and Bm is the motor viscous damping coefficient

(1)

where QL is the load flow rate; Cd is the discharge coefficient; w is the width of each port; d. is the spool displacement; p is the fluid mass density; PL is the supply pressure; and PL is the load pressure.

From (1), the flow rate through the orifice depends on the value of the load pressure which is time varying during a system transient period. In addition, the fluid bulk modulus, ~, in (3) is a pressure-dependent parameter. The pressure dependence of the bulk modulus is difficult to measure because it varies with the volume of the entrained air in the fluid. Equations (1) through (4) can be rewritten as a set of "linear" state-space equations with pressure-dependent parameters as follows:

where XI == Nm , X2 == PL, and Fig. 1. Computer-controlled electrohydraulic servosystem

arl = - Bm/J, ar2 =Dm/J, ail = - 4 Dm ~qx~)N, a22 = - 4 Cc ~Cx~)N,

Since the valve serves as a control component, it is usually designed to have much higher speed of response than the other system components. Hence, the valve dynamics are neglected, and a proportional relation between the valve input current, I, and the spool displacement, d., is assumed:

P; = 4 Cd W Kv ~CX~)(P'1X~)IIlI(VtPIIl)

(5b) (Se) (5d) (Se) (Sf)

Hereafter, a greek parameter indicates that it depends on the load pressure. Taking the Z transform of (Sa) gives the following discrete model:

(2)

Application of the continuity equation to the motor chambers yields (3)

where

where ~ is the fluid bulk modulus; Dm is the volumetric displacement of motor; Nm is the motor angular velocity; Cc is the motor leakage coefficient; V, is the total volume of fluid under compression (includes the volumes of the connecting lines and dead volumes between the servovalve and motor). The ideal torque generated by the pressurized fluid is Dm PLo However, there are torque losses due to viscous and Coulomb friction, and stiction. Only viscous friction is considered in this study. A torque balance on the motor shaft results in the following equation: and h denotes the sampling time interval; Al and A2 are the system pressure-dependent poles in the continuous-time domain.

(4)

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the gains are adjusted in relation to the estimated value of the load pressure. The control law is designed based on the following model:

Equations (6) are equivalent to al2 ] [xll + (l22 I X2

[~Il ~

u(t) (7a) I

~12l [:~1 + [~Il u(t) (In

if ~I

=~I

~

I

(l11 + (In = (l11 + (l22 (l11(ln =al2a21 + (l11(l22 - (l12(l21

(7b) (7c) (7d)

~1(l22 + a12~2 = ~la22 + a12~

(7e)

(8)

I

The pressure dependence of (In, (l22, ~I' and ~2

can be derived in a closed fonn with uncertainties in the values of the discharge coefficient and the fluid bulk modulus. However, the fonnulations are not in useful forms for parameter estimation. Further discussion on the estimation of the pressure-dependent system parameters is given in a companion paper (Un et ai, 1993).

Fig. 2. A block diagram of state variable feedback via observer and unity output feedback control

3. STATE VARIABLE FEEDBACK VIA OBSERVER AND UNITY OUTPUT FEEDBACK CONTROL The use of state variable feedback control based on pole assignment is a well-known technique to shape system dynamic behavior. The technique was enhanced by introducing a co-state in the feed forward path along with an unity output feedback (Christensen et ai, 1986) in order to guarantee tracking control accuracy. Fig. 3. A block diagram of state variable feedback via observer and unity output feedback control with pressure-dependent gains

Figure 2 shows a block diagram of Christensen's controller implemented with an observer. The control gains, g's, are fixed and detcnnined based on a set of desired closed-loop pole locations.

For convenience, the observer in Fig. 3 is neglected and will be considered later. The control input is then detennined by

However, for a system with signal-dependent parameters controlled by a linear feedback controller, the closed-loop pole locations also would be signal dependent. As tbe signal has a large variation during the control process, the pole locations also vary and system stability is not guaranteed.

(9a)

where g(X2) =[gl(X2), gz(X2), g3(X2)]

T

=[XI(t), X2(t), e(t)] e(t+l) = e(t) + Xld(t) - XI(t)

If the gains of a linear controller arc adjusted in

~(t)

relation to changes of the system parameters, the signal-dependent dynamics can be compensated. The design of such a controller can be achieved by extending the design methodology of Christensen.

T

(9b) (9c) (10)

The introduction of the co-state e(t) guarantees that the system output XI (t) will ultimately track the desired output xlit), so long as the closed-

3.1. Controller with Signal-Dependent Gains

loop system is stable or the feedforward loop reaches steady state.

Figure 3 shows Christenscn's controller, except

Substituting (9a) into (8) and combining the

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........ " ,.." (XI(t+ 1)-(lIIXI(t)-~lu(t)-al2X2(t»)

result with (10) gives 3,(t+ I) =

A x(t) + 12 Xld(t)

(lla)

where &0 denotes the observer gain which also will be adjusted at each sampling instant in order to compensate for the variation of the observer pole.

(lIb)

For comparison, Christensen's controller with fixed gains is considered and designed based on the following linear model:

where

_

[~U-!Igl :12-~1&2 -~11&31

A - a2J-jhgl (l22-ih&2 -13zgJ -I 0 h =[ 0, 0, I ]T

(14)

(llc)

Let the desired closed-loop poles be the roots of the following characteristic equation: The control gains are determined using (13) with the pressure-dependent (l'S and p's replaced by their corresponding constant a's and b's.

(12) The control gains become: (Ba)

4. AN EXAMPLE

where

The open-loop system model parameters in (8) and (15) were determined using experimental data obtained by subjecting an actual servosystem to a pseudo-random input with an uniform distribution. The linear constant-parameter model was achieved offline using the recursive leastsquares estimator and the model with pressuredependent parameters (called the proposed model) was achieved offline using an iterative estimator. The results for a sampling interval of 20 msec and a supply pressure of 500 psig are summarized below (Lin, 1992):

(Bb) (Bc)

(Bd) ..-2

Y2(X2)

=~I

(Be)

'Y3(X2) = ~I(a21~1-HX22~2) - fu
........ 2

,..,..

.......

911(X2) = (l11-H;1(l11+(2aI2a21-H;2)(l11+ aJ:2a2I(a22-H;I)+;3 ~2

........................ 2

(13g) ........

Linear Model (see (15»

........

912(X2) = ~(l11+(l11(l22+(l2z-H;1«(l11+«(l22)+ alza21-H;2) 1112 (13h) 921(X2) = 912 a211al2 (l3i) -3

-2

..-..-

-

9z2(X2) = (l22-H; I(l22+(2aI2a21 +;2)(l22+ l112a21(al1-H;I)+;3 "",2

,.. ,..

........,.. ,..

,..

A,...2,.,.,...2........

(131) (Bm) A

(16b) (16c)

a21 = -3.63 (l3j)

931(X2) = -(l11+{l-H;I)(l11-(aI2a21+1-H;1-H;2) (13k) 932(X2) = -(al1+a22+I+;I)a12 S33 = 1+;1-H;2+;3

(16a)

all = 0.856 a12 = 0.044 a22 = 0.802 '" ,..bl = 1.24

(16d)

b2= 53.1

(16f)

(16e)

Prowsed Model (see (8»

"-

5(x2) = (aI2~2+~IXaI2~-a21~1 +(lII~I~).......,.,.

",..2 "" ,,2 ",..

al1 = 0.87'Ps(x2)+(0.40+0.oollx~)'PL(x2) (17a)

(l22~I(2a12~-a21~1 +~1~)+ ,..lA ........

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

~1 fua22«(l22-(l11)

(13n)

al2 = 0.045

(l7b)

a21 = -3.79

(17c)

The measurement of X2 for the calculation of the control input is not required if an observer is

a22 = (1.03-0.00~x~)'PS(X2) +

used:

'" PI = (1.32-0.003Ix~)'PS(X2) +

(1.08-0.oollx~)'PL(x2)

(8.73-0.0231x~)'PL(x2)

(l7d) (17e)

'"

~ = (61.8-0·13Ix~)'Ps(x2) + (l20.2-0.231x~)'PL(x2)

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(17f)

(a) designed based on the linear model

where

400

'l'S(X2) =~(x2)/(JJs(X2)+IlL(X2» (17g) 'l'L(X2) =~L(X2)/(JJs(X2)+IlL(X2» (17h) (342.21x21)/342.2, if 0 S Ix21 < 342.2

~={

IlL ={

o o

(rpm

200

, otherwise

qx21-74)/(Ps-74), if 74 < Ix21

measured response

Xlf

s

simulation response

(17i) Ps

(the linear model as the nominal system)

, otherwise

O~~--~-----r----~----~

o

(17j)

2

time (sec)

4

(b) designed based on the proposed model

Two cases of tracking controls were considered to illustrate the feedback control behavior. The goal was for the closed-loop system motor speed (simulation and measured) response to follow the desired response, Xld, which was assumed to be an exponential function with a different final steady-state level for each case.

400

measured response

Xlf (rpm

200 simulation

res~nse

(the proposed model as the nominal system)

A triple exponential digital filter (Le., three firstorder digital filters in cascade) was used to reduce the effects due to measurement noise. The filtered signal, xlf (not shown in Fig. 3), was used for the calculations in (9), (10), and (14), instead of the measured one.

O~~--~-----r----~----~

o

2

time (sec)

4

Fig. 4. Measured and simulation responses for the closed-loop system, Case 1

Closed-loop simulation results based on the linear model (Fig. 4(a) and 5(a» and the proposed model (Fig. 4(b) and 5(b» are plotted as dashed lines. The poles were assigned at 0.6 ± O.lj and 0.6, the observer pole was assigned at 0.85, and the cut-off frequency of the filter was set at 10 Hz. The determination of pole locations was based on the classical linear design concept (Frankin et al., 1980).

(a) designed based on the linear model 150

100 xlf

(rpm) 50

The conditions used for the simulations were implemented experimentally. The digital controllers were implemented on an 80386-based computer. Experimental results showed that the controller designed based on the linear model and proposed model worked equally well for the operating condition of Case 1. Typical measured responses are plotted in Fig. 4 in solid lines. For this case, the simulation responses are in close agreement the measured responses.

simulation response

(the linear model as the nominal system) O~~--~-----r-----'----~

o

2

time (sec)

4

(b) designed based on the proposed model

150 measured response

As the operating condition changed from Case 1 to Case 2, experimental results showed that the fixed-gain linear controller was no longer adequate. Further, the closed-loop system stability deteriorated. The system dynamic behavior for the operating condition of Case 2 is not well described by the linear model. As shown in Fig. 5(a), the simulation response of the closed-loop system based on the linear open-loop system model cannot predict the occurrence of the

O~~--'-----~-----r-----'

o

2

time (sec)

Fig. 5. Measured and simulation responses for the closed-loop system, Case 2

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4

system instability. It is concluded that the modeling error must be considered in order to adjust the control gains and to achieve good system stability.

of Dynamic Systems, Measurement, and Control, Vo!. 107, June, 145-150. Frankin, G.F. and J.D. Powell (1980). Digital control of dynamic systems. Addison-Wesley Publishing Company, Inc. Lin, K.C. (1992). An approach 10 modeling and identification of systems with signal-dependent parameters. Ph.D. dissertation, MAE Department, Oklahoma State University. Lin, K.C. and K.N. Reid (1993). An approach 10 modeling and identification of systems with signal-dependent parameters. to be presented at the Triennial World Congress ofIFAC. Merrit, H.E. (1967). Hydraulic control systems, John Wiley & Sons, Inc .. Sung, D.J.T. and T.T. Lee (1987). Model reference adaptive control of a solenoid valve controlled hydraulic system. Int. J. System Sci., Vo!. 18, No. 11,2065-2091. Vaughan, N.D. and I.M. Whiting (1986). Microprocessor control applied to a nonlinear electrohydraulic position control system. 7th International Fluid Power Symposium, Bath, England, 16-18, September, 187-197. Watton, J. (1990). A digital compensator design for electrohydraulic single-rod cylinder position control systems. ASME Journal of Dynamic Systems, Measurement, and Control, Vo!. 112, September, 403-409.

On the other hand, the system stability and steady-state accuracy were significantly improved for the controller designed based on the proposed open-loop system model when the above change of operating condition occurred. A typical measured response is oscillatory but tracks around the desired response. This behavior occurs because the open-loop system is very lightly damped for the operating condition of Case 2, and the mter dynamics were neglected in the design of the controller. Also, shown in Figure 5(b) is the simulation response of the feedback system designed based on the proposed open-loop system model. Good prediction of the oscillatory response is achieved even though there is a significantly initial delay between the measured and the simulation responses due to motor stiction.

5. CONCLUSIONS A linear model with pressure-dependent parameters can be developed to represent the non linear dynamic behavior of an open-loop electrohydraulic servosystem with a reasonable degree of accuracy. This open-loop model can be used 10 design a feedback control system based on a classical 1;'1ear design concept. The resulting controller has signal dependent gains, and the ability to compensate for system nonlinearities such that the system behaves well over a wide range of operating conditions.

6. REFERENCES Christensen, G.K., F. Conrad, N.E. Hansen, and E. Trostmann (1986). Design of hydraulic servo with observer and unity output feedback 7th International Fluid Power Symposium, 16-18, September, 199-209. Daley, S. (1987). Application of a fast selftuning control algorithm to a hydraulic test rig. Proc. I. Mech. E., Vo!. 201, No. C4, 285-295. Edge, K.A. and K.R.A. Figueredo (1987). An adaptively controller electrohydraulic servomechanism, Part 2: implementation. Proc. I. Mech. E., Vo!. 201, No. 83, 181-189. Finney, J.M., A. de Pennington, and G.S. Gill (1985). A pole-assignment controller for an electrohydraulic cylinder drive. ASME Journal

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