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Optimum response design guides for electrohydraulic cylinder control systems
Optimum response design guides for electrohydraulic cylinder control systems J. Watton School of Engineering, Division of Mechanical of Wales College of CardifA Card&$ UK
Engineering
& Energy
Studies,
University
The dynamic performance of servovalve-controlled linear actuators is studied for the systems used in practice. Both open-loop control and closed-loop control are considered, and use is made of the linearized transfer functions. Some design rules are established from the linearized theory, and the optimum response is then established by further computer analysis of the exact system theory. Design curves are then presented to enable optimum velocity and position response to be produced for a range of variation in the system parameters. Keywords: fluid
power, optimum response
Introduction In modern fluid power control systems design, more attention is being given to optimum dynamic performance of cylinders and motors used for position and speed control. The author has been involved in a number of projects ranging from agricultural machines to systems in the forging industry, and all require a consideration of operating dynamics. Inevitably, a full computer simulation is undertaken, but in every case a preliminary linearized analysis of the system has provided useful design information. Since there are a number of electrohydraulic control systems in common use, it is timely to review the linearized solutions using data from the exact simulations. This should then give the systems designer a standard set of transfer functions that are applicable to a preliminary response and stability study. Transfer function techniques will always play a useful role in the understanding of hydraulic systems dynamic performance. Systems with nonlinear components can be assessed by using this approach, particularly with regard to stability, 1-5 and this can be invaluable in considering the setting up of a complicated exact computer simulation.
Address reprint requests to Dr. Watton at the School of Engineering, Division of Mechanical Engineering and Energy Studies, University of Wales, College of Cardiff, P.O. Box 917, Cardiff CF2 IXH, UK. Received
598
29 January
Appl.
1990; accepted
Math. Modelling,
9 May
1990
1990, Vol. 14, November
If it is assumed that a transfer function approach is adequate for a preliminary design study, then it would seem desirable to be able to determine what the optimum form of the transfer function should be. If standard forms could be established, then it would be simply a matter of selecting the component parameters to match the optimum values required. The ensuing analyses therefore attempt to bring together a range of transfer functions that may be used for design purposes and for commonly occurring servovalve/actuator combinations. In particular, the following combinations are considered: 1. Single-rod cylinder, extending and retracting, openloop velocity control 2. Single-rod cylinder, extending and retracting, closedloop velocity control 3. Single-rod cylinder, closed-loop position control. The piston is_. assumed __ to be supporting a vertical load acting such that it always attempts to retract the piston, that is, it always acts in the same direction. For open-loop control systems a critically lapped servovalve spool is assumed, and for closed-loop control systems a symmetrically underlapped servovalve spool is always assumed. The latter assumption is more useful for closed-loop analysis, since a small amount of spool underlap often occurs in practice. It is also an easy matter to reduce the underlap to zero to simulate a critically lapped spool, should this be desirable. The servovalve flow equations may therefore be written as follows:
0 1990 Butterworth-Heinemann
Optimum critically
lapped
a
Ips
symmetrically underlapped
response design for control systems: J. Watton Q, = kjm
(1)
Q2 = k&a
(2)
Ql = kf(i, + i)m Q2 = kf(i, + i>fl
- kf(i, + i)e - kf(i, - i)m
(3) (4)
dynamic effects are evident. Further details on servovalve dynamic modelling may be found in Refs. 6-13, should this refinement be necessary. The line and load dynamic equations for the extending case are therefore
Servovalve dynamics are neglected, and it is also assumed that the dominant dynamic contribution is made by load inertia and line compressibility. In all cases the volumes on each side of the system are assumed to be equal and constant during the period when
Q,=A,U+CclP,
(5) (6) (7)
parameter to give a 4.6% overshoot. A generalized approach is used throughout, using common definitions for dynamical elements as well as system gains. This is done for open-loop control by assuming that a fictitious input into the system such that the open-loop transfer function may also be defined in terms of the servoamplifier gain and, in this case, the fictitious feedback amplifier gain. Hence for cylinder open-loop velocity control the input is I!_/,,the demand velocity. Figure I shows the system for which it will be deduced that two conditions are possible, extending or retracting. The steady-state pressures in each case are as follows:
A variety of techniques such as analog simulation, digital simulation, and linearization methods have been used to study electrohydraulic control systems.‘,‘4-1’ The author has attempted to generalize some transfer functions,‘,20*21 but there seems to be little information on optimum forms. If a transfer function approach is adopted, then the relatively simple forms suggest that it should be possible to lead to an optimum design using classical second-order definitions or the ITAE response criteria.‘,2z-24 Therefore in response to a step input, the overshoot will be 4.6% for a second-order system and 2% for a third-order system. In this work the ITAE criterion is used where possible, and the final design is obtained by considering the exact solution, via digital simulation, and adjusting the appropriate
p
Extending:
,
= (1 + ~2R~s
p2
=
p
I
= Y2U + nps (1 + Y3)
m-Ys
-
(8)
(1 + r3)
(1 + r3)
Retracting:
(Y
p2
=
(Y3
-
nps
(9)
(1 + Y3>
where
Therefore the steady-state are given by and
F=F
velocities in either direction
(10) PA2
u=
Urefa
Uref
-
&
y
(11)
2
Appl. Math. Modelling,
1990, Vol. 14, November
599
Optimum
response design for control systems: J. Watton
and 2(Y - F,
extending a =
(1 + r3)
-
retracting a = Figure 1.
Open-loop
speed control of a single-rod
cylinder
J
2(1 F, (, ++ y3)
(12)
Considering the linearized analysis of equations (I), (2), (5), (6), and (7) and incorporating (8) and (9) lead to a generalized transfer function that is applicable for the extending and retracting cases and is given by
[1+F*w] +FXy(l + y)+ S'Xy%z I+s a(y+ r3) + XY +r3) (1 +r3) +r3) (1 +y3)a [(1 1 (I K,a
(13)
S = sCR,,
Considering transfer function (13), it follows that the linearized dynamic response will be the same in both directions provided that the a values are equal. From equation (12) this is possible, provided that
y-1
F=T-
(-J=
J -
l+Y
(15)
1 + y3
It follows from equation (11) that the steady-state velocities in both directions are then equal. Further consideration of the transfer function shows that it is now necessary only to specify the “time constant ratio” X to satisfy the response overshoot requirement for each area ratio y. A guide to this selection is established by considering the double-rod case, y = 1 and F = 0, where the transfer function simplifies to second-order form. The solution for a step response overshoot of 4.6% gives X = 3.92, and the exact value of the nonlinear system is obtained from the digital simulation. Model 1 in the appendix is used, and the solution is shown in Figure 2. Generally, the actual parameter X to be used is lower than that predicted by linearized theory, since the ex-
(14)
act solution gives more damping than the linearized solutions at the same operating conditions. This is particularly true in this case because the exact solution is valid at maximum rod velocity only. A set of transient responses is shown in Figure 3 for area ratios y = 1, 4,2, at optimum load force, and for the extending case. The responses for the retracting case are identical except that the pressure roles are reversed, that is, Pz appears like P, and vice versa. The velocity responses are identical in magnitude. Single-rod
cylinder,
closed-loop
speed control
Figure 4 shows the closed-loop
steady-state
0.5
1
pressures
\
Y =I
,_----_
‘\.,k_ +_______ L<_----
system in which the for both extending and retract-
2
05 -p
i
Figure 3. Some optimized transient responses for y = 1, +, 2, F = (y - 1112 for the open-loop system
1L 1
Figure 2. Optimum withF= (y1)/2
600
Appl.
Math.
1.2
1.4 area
1.6 rat10
1.8
2
Y
X for the open-loop
speed control system Figure 4.
Modelling,
1990,
Vol. 14, November
Closed-loop
speed control of a single-rod cylinder
Optimum
ing cases are identical with the open-loop system previously discussed. The steady-state velocity now depends upon the gain, KU, as well as the load force F. Hence the steady-state servovalve current is influenced by these parameters, and the linearized servovalve resistances are different from R, used in the purely open-loop system. For this reason, transfer function (13) cannot be used for closedloop analysis. It is an easy matter to show that the steady-state current, i,, is given by ~,ef 1,
=
iref = G, ff,i
1 + K,,a’
-=
Ud,
ud=$
U a
K
aud(s)
1 + K,a +-
s
and therefore u -= r/Cl
velocity,
U, is given by
K,a 1 + K,,a
(17)
where a is as defined in equation (12) and is different for the extending and retracting cases. Linearizing the system equations now gives the following closed-loop transfer function:
1
+
s(1 + K,aMy + r3) (1 + r3)
a(1 + K,a)2(y + y3) +
(1 + r3)
the steady-state
(16)
I‘
6Ws)
response design for control systems: J. Watton
Again, as for the open-loop system, the linearized dynamic response is identical in either direction, provided that equation (15) is satisfied. This is also an easy matter to show that for the double-rod case, y = 1 and F = 0, the resulting second-order transfer function satisfies the optimum response criterion provided that X = 3.92(1 + KU)3. This again is used as a design guide when considering the exact simulation results. In this case, Model 2 in the appendix is used, and the requirement of a 4.6% overshoot in response to a step input gives the results shown in Figure 5 for a range of areas ratios, y, and gain, K,,. The results presented in Figure 5 show how the design parameter X must increase as the system gain, K,,, is increased. All the values fall below the linearized solution for y = 1, but of course they converge as K,, is decreased such that the dynamic velocity fluctuation from the initial condition is reduced. Some transient responses are shown in Figure 6 for an area ratio y = 4 and different values of the system gain. It will be observed from Figure 6 that cavitation may occur as the system gain is increased beyond a
I +S2Xy(l +y)+
1
XY a(1 + K,a)
S3Xy2a(l + K,,a)
(1 + r3)
(1 + Y’)
value typically K,, = 1. Similar conclusions y = 1 and y = 2. Single-rod
cylinder,
closed-loop
(18)
hold for
position control
Figure
7 illustrates the closed-loop position control system utilizing a transducer with gain HP. This control system using a symmetrically underlapped spool has been studied for the variable volume case,2’ and the concept of a steady-state position error that varies with load force was introduced. It has been shown that a zero steady-state error, obtained from the exact solution, is only zero provided that
(19)
Figure 6. Some optimized transient responses for y = 4, F = a for the closed-loop velocity control of a cylinder
I I
I llllill
0.1
Figuy_e5. Optimum andF= (y1)/2
i
’ Xfor
Ku
,
Iilllil IO
the closed-loop speed control system Figure 7.
Closed-loop
Appl. Math. Modelling,
position control of a single-rod
1990, Vol. 14, November
cylinder
601
Optimum
response design for control systems: J. Watton
This is remarkably the same requirement as for dynamic response symmetry for the speed control systems previously analyzed. Therefore the generalized closed-loop transfer function, which always indicates
-SYY(S) = @d(S)
,
+
St1
+
r2)
+
2( 1 + Y)K,
a zero position error, is truly valid only for one load force. Since the steady-state pressures are always equal to half the supply pressure, this transfer function simplifies to the following form:
1 S2XY ?XY (1 + Y)K, + 2(1 + Y)&
(20)
where FE
l
y-
2
K
’
P,
GJfpKf
=
A2
’
S = sCR,,,
J 1’
Ep = K&R,,
Yd
=
x = 0.356(1 + Y2)
P (1
+
Y
Y)
=
yd
Yref =
Yref
=m
(21)
k&,
(22)
(23)
Therefore, a particular area ratio is associated with a unique system gain and time constant ratio parameter. The computer simulation of the exact system equations is again used to reassess the linearized analysis prediction, and the proposal is that the parameter X be as defined in (23) and the gain K,, be modified to give the transient response overshoot of 4.6%. This can again be done for a range of area ratios y using Model 3 given in the appendix. However, since position control is perhaps more common than velocity control, the effect of position demand on the optimum performance has also been studied. Figure 8 therefore shows the system gain required for nondimensional demanded positions of yd = I, 2, 4, 8, where the reference position is given by Ld
I,
VdIHp
It is now possible to match the third-order ITA criterion to transfer function (20), and the solution is K = 0.266(1 + Y2)
R
F
The effect of an increased demanded position is to slightly increase the gain required to give the desired optimum response. However, retaining the idealized value given in (23) will result in an adequate response for a wide range of operating conditions. A typical set of transient responses are shown in Figure 9 for an area ratio y = 4 and for a range of demanded positions. The effect of an increased demanded position is to probably induce cavitation as the pressure Pi (when extending) falls to zero. This tendency to cavitate is removed as the area ratio is decreased toward Y = 1, the double-rod case. Conclusions The linearized analysis technique was found to be a useful guide to determining an adequate optimum response for the three configurations considered. It was found in all cases that symmetry of dynamic resnonse mav be achieved by matching the supply pressure to the load. This unique setting for both ve-
(24)
2
9
L
T&=8
B
---..__
,’
2
,_y--__
, __--___1__ IC
0.1
II III1
Ol
5t/t
-__4 -.2 -----_ , ..- l,neor,sed analysts -
I
!
I 1.2
I
I
1
I I,
I
I I I 1.6 I.4 area ratm Y
I
I
i
11
I
I.8
I]
Appl.
Math.
Modelling,
1990,
1
I
I 2
Figure 8. Optimized system gain from closed-loop control and for 7 = (y - 1)/2, X = 0.35611 + y*)
602
10
position
Vol. 14, November
Figure 9. Some optimized transient responses for y = j, F = t and for closed-loop position control of a cylinder
Optimum
locity and position control is achieved by using F = (y - 1)/2 or P, = 2Flrod area. It follows that the steady-state velocities are identical in both directions for the open-loop and closed-loop velocity control systems. For the range of area ratios considered, y = 1-2, the design parameter X is lower than the linearized solution for the open-loop and the closed-loop velocity control system. This follows an anticipated trend, since small-signal linearization inherently suggests lower damping than the large signal practical case.’ Hence a lower X-value, suggesting a lower inertia to give more overshoot, is required to satisfy the response criterion. For the closed-loop velocity control system a working design rule would be X = 3. I( 1 + KIJZ and adequately embraces the area ratios of interest. The optimum ITAE criterion for position control requires specific values of both the-time constant ratio X and the nondimensional gain KP: The design approach adopted is to assume the optimum X required, that is, X = 0.356(1 + r2)lr and then select the gain term from the exact simulation. This gives a solution again always above that suggested by a linearized analysis, the increase depending upon the magnitude of the movement required. The trend corresponds to the linearized solution and for the demand positions studied, &yref = 1-8, a satisfactory design rule would be K, = 0.29(1 + y’)l(l + $. Large pressure fluctuations were found to occur in many instances for each of the three control systems. For the open-loop velocity control system, maximum and minimum pressures were relatively insensitive to the area ratio selected, and in all the cases studied these were close to supply pressure and zero gauge pressure respectively. Similar comments hold true for closedloop velocity control, particularly as the system gain is increased. Closed-loop position control is more tolerant of large pressure fluctuations, but the limits of supply pressure and zero gauge pressure will inevitably be reached for large demanded changes in position. In all cases, therefore, it might be worth considering pressure feedback and reassessing the response criteria.
ud u ref
V vd
X Y Yd
Yref P Y 7
References I 2
3 4
5
F, F G, f& HU i, T lref 1,
kf
-
KP’
K/J
Watton, J. Fluid Power alog and Microcomputer
Systems: Modeling, Simulation, AnControl. Prentice-Hall, Englewood
Cliffs, N.J., 1989 Martin, K. F. Stability and step response of a hydraulic servo with special reference to unsymmetrical oil volume conditions. J. Mech. Engrg. Sci. 1970, U(5), 331-338 Urata, E. Influence of compressibility of oil on the step response of a hydraulic servomechanism-the response under inertia loading. Bull. JSME 1982, 25(203), 797-803 Takahashi, K. and Takahashi, Y. Dynamic characteristics of a spool valve controlled servomotor with a non-symmetrical cylinder. Bull. JSME 1980, 23(10), 1155-l 162 Goodwin, A. B. Fluid Power Svstems. Macmillan. New York, I979
6
Watton, J. The effect of drain orifice damping on the performance characteristics of a servovalve flapper/nozzle stage. J.
7
Zaborszky, J. and Harrington, H. J. Generalised charts of the effects of non-linearities in 2 stage electrohydraulic control valves. AIEE Trans. 1958, 76, 401-408 Feng, T. Y. Static and dynamic control characteristics of flapper-nozzle valves. ASME 1. Basic Engrg. 1959, 81, 275-284 Merritt, H. E. Hydrclulic Control System. John Wiley and Sons, New York, 1967 Nikiforuk, P. N., Ukrainetz, P. R. and Tsai, S. C. Detailed analysis of a two-stage four-way electrohydraulic flow-control valve. J. Mech. Engrg. Sci. 1969, 11(2), 168-174 Martin, D. J. and Burrows, C. R. The dynamic characteristics of an electrohydraulic servovalve. ASMEJ. Dynamic Systems, Measurement and Contra/ Dec. 1976, 395-406 de Pennington, A., ‘t Mannetje, J. J. and Bell, R. The modelling of electrohydraulic control valves and its influence on the design of electrohydraulic drives. J. Mech. Engrg. Sci. 1974,
Dynamic
8 9 IO
linearized parameter cylinder areas fluid capacitance (V/p) cylinder load force and nondimensional value (F/P,AJ servoamplifier gain position transducer gain velocity transducer gain servovalve current and nondimensional value (i/i,,) reference servovalve current servovalve current at the spool underlap boundary servovalve flow constant position gain and nondimensional value (KZ’R)
velocity gain load inductance (m/AIA2) load mass pressures on either side of the system supply pressure to the servovalve flow rates on either side of the actuator servovalve linearized resistances Laplace operator and nondimensional value (SW cylinder velocity demand velocity reference velocity volume on either side of actuator control system demand voltage time constant ratio (L/CR2) cylinder position demanded cylinder position reference cylinder position fluid bulk modulus cylinder area ratio (Al/A,) simulation reference time constant
u
Notation
:I, A* c
response design for control systems: J. Watton
II 12
I3 14 I5 16 17
Appl.
Systems,
Measurement
and Control
1987,109,19-23
16(3), 196-204 Transfer Function for Dowty-Moog
Servovalves. Available from the manufacturer. Watton, J. Fluid power design by CAD. Professional Engrg. March 1989, 33-34 Backe, W. and Hoffmann, W. DSH programme for digital simulation of hydraulic systems. Paper Cl, BHRA 6th International Fluid Power Symposium, Cambridge, UK, 1981 Kinoglu, F. et al. Streamlining hydraulic circuit designs with computer aid. Comput. Mech. Engrg. 1982, l(2), 21-26 Vilenius, M. J. The application of sensitivity analysis to electrohydraulic position servos. J. Dynamic Systems, Measurement and Control 1983, 105, 77-82
Math. Modelling,
1990, Vol. 14, November
603
Optimum 18 19 20
response design for control systems: J. Watton
Karnopp, D. Bond graph models for fluid dynamic systems. J. Dynamic Systems Measurement and Control 1972, Series G, 94(3), 222-229 Thoma, J. U. Simulation by Bondgraphs. Springer-Vet-lag, New York, 1989 Watton, J. The generalised response of servovalve-controlled, single-rod, linear actuators and the influence of transmission line dynamics. .I. Dynamic Systems, Measurement and Control 1984, 106, 157-162
21
22 23
Watton, J. The effect of servovalve underlap on the accuracy and dynamic response of single-rod actuator position control systems. J. Fluid Control 1988, B(3), 7-24 Ogata, K. Modern Control Engineering. Prentice-Hall, New York, 1970 Graham, D. and Lathrop, R. C. The synthesis of optimum response criteria and standard forms. AZEE Trans. Part II 1953,
x=&g 0
Owing to the matched load such that F = FIP,A2 = 1)/2, the retracting case gives identical velocity responses and pressure responses with the roles of P, and P2 interchanged. Single-rod cylinder, closed-loop velocity control (Model 2) extending: (y -
72, 273-288
24
Baz, A. Optimisation of the dynamics of pressure compensated axial piston pumps. J. Fluid Control 1983, 15, 64-81
Appendix: System exact equations
dp,
The appropriate servovalve flow equations (1) and (2) or (3) and (4) are combined with the actuator flow continuity and momentum equations (5), (6), and (7). These equations are then nondimensionalized in a manner appropriate to the system being studied, the reference values being
jY=-__
F
(y-
PsAz
dt/7 = K,( 1 - @V/2(1 - P,) - yu dp, -z dtl7
i? - K,(l
- u)m
1) 2
Single-rod cylinder-open loop velocity control (Model 1) extending:
Again design symmetry results in the retracting case giving similar conclusions as for the open-loop velocity control system. Note that for the closed-loop system the reference time, 7, is a function of the system gain, K,. Single-rod cylinder, closed loop position control (Model 3):
-
$i
= ti
Z,(l + i)m
$T
sign (1 - P,) - Z,( 1 - ;)fi]
[
= ti
Zr(l + t)fl
- Zz(l - i)U/FFJ
sign (1 - P2)
[
- $9
1
+ V
df? dt/r 2 dtly
Yref
Zl?
LLJJ
sCP A2
CR,
I
604
Appl.
Math. Modelling,
z2t
1990,
Vol. 14, November
Engineering
& Energy
Studies,
University
The dynamic performance of servovalve-controlled linear actuators is studied for the systems used in practice. Both open-loop control and closed-loop control are considered, and use is made of the linearized transfer functions. Some design rules are established from the linearized theory, and the optimum response is then established by further computer analysis of the exact system theory. Design curves are then presented to enable optimum velocity and position response to be produced for a range of variation in the system parameters. Keywords: fluid
power, optimum response
Introduction In modern fluid power control systems design, more attention is being given to optimum dynamic performance of cylinders and motors used for position and speed control. The author has been involved in a number of projects ranging from agricultural machines to systems in the forging industry, and all require a consideration of operating dynamics. Inevitably, a full computer simulation is undertaken, but in every case a preliminary linearized analysis of the system has provided useful design information. Since there are a number of electrohydraulic control systems in common use, it is timely to review the linearized solutions using data from the exact simulations. This should then give the systems designer a standard set of transfer functions that are applicable to a preliminary response and stability study. Transfer function techniques will always play a useful role in the understanding of hydraulic systems dynamic performance. Systems with nonlinear components can be assessed by using this approach, particularly with regard to stability, 1-5 and this can be invaluable in considering the setting up of a complicated exact computer simulation.
Address reprint requests to Dr. Watton at the School of Engineering, Division of Mechanical Engineering and Energy Studies, University of Wales, College of Cardiff, P.O. Box 917, Cardiff CF2 IXH, UK. Received
598
29 January
Appl.
1990; accepted
Math. Modelling,
9 May
1990
1990, Vol. 14, November
If it is assumed that a transfer function approach is adequate for a preliminary design study, then it would seem desirable to be able to determine what the optimum form of the transfer function should be. If standard forms could be established, then it would be simply a matter of selecting the component parameters to match the optimum values required. The ensuing analyses therefore attempt to bring together a range of transfer functions that may be used for design purposes and for commonly occurring servovalve/actuator combinations. In particular, the following combinations are considered: 1. Single-rod cylinder, extending and retracting, openloop velocity control 2. Single-rod cylinder, extending and retracting, closedloop velocity control 3. Single-rod cylinder, closed-loop position control. The piston is_. assumed __ to be supporting a vertical load acting such that it always attempts to retract the piston, that is, it always acts in the same direction. For open-loop control systems a critically lapped servovalve spool is assumed, and for closed-loop control systems a symmetrically underlapped servovalve spool is always assumed. The latter assumption is more useful for closed-loop analysis, since a small amount of spool underlap often occurs in practice. It is also an easy matter to reduce the underlap to zero to simulate a critically lapped spool, should this be desirable. The servovalve flow equations may therefore be written as follows:
0 1990 Butterworth-Heinemann
Optimum critically
lapped
a
Ips
symmetrically underlapped
response design for control systems: J. Watton Q, = kjm
(1)
Q2 = k&a
(2)
Ql = kf(i, + i)m Q2 = kf(i, + i>fl
- kf(i, + i)e - kf(i, - i)m
(3) (4)
dynamic effects are evident. Further details on servovalve dynamic modelling may be found in Refs. 6-13, should this refinement be necessary. The line and load dynamic equations for the extending case are therefore
Servovalve dynamics are neglected, and it is also assumed that the dominant dynamic contribution is made by load inertia and line compressibility. In all cases the volumes on each side of the system are assumed to be equal and constant during the period when
Q,=A,U+CclP,
(5) (6) (7)
parameter to give a 4.6% overshoot. A generalized approach is used throughout, using common definitions for dynamical elements as well as system gains. This is done for open-loop control by assuming that a fictitious input into the system such that the open-loop transfer function may also be defined in terms of the servoamplifier gain and, in this case, the fictitious feedback amplifier gain. Hence for cylinder open-loop velocity control the input is I!_/,,the demand velocity. Figure I shows the system for which it will be deduced that two conditions are possible, extending or retracting. The steady-state pressures in each case are as follows:
A variety of techniques such as analog simulation, digital simulation, and linearization methods have been used to study electrohydraulic control systems.‘,‘4-1’ The author has attempted to generalize some transfer functions,‘,20*21 but there seems to be little information on optimum forms. If a transfer function approach is adopted, then the relatively simple forms suggest that it should be possible to lead to an optimum design using classical second-order definitions or the ITAE response criteria.‘,2z-24 Therefore in response to a step input, the overshoot will be 4.6% for a second-order system and 2% for a third-order system. In this work the ITAE criterion is used where possible, and the final design is obtained by considering the exact solution, via digital simulation, and adjusting the appropriate
p
Extending:
,
= (1 + ~2R~s
p2
=
p
I
= Y2U + nps (1 + Y3)
m-Ys
-
(8)
(1 + r3)
(1 + r3)
Retracting:
(Y
p2
=
(Y3
-
nps
(9)
(1 + Y3>
where
Therefore the steady-state are given by and
F=F
velocities in either direction
(10) PA2
u=
Urefa
Uref
-
&
y
(11)
2
Appl. Math. Modelling,
1990, Vol. 14, November
599
Optimum
response design for control systems: J. Watton
and 2(Y - F,
extending a =
(1 + r3)
-
retracting a = Figure 1.
Open-loop
speed control of a single-rod
cylinder
J
2(1 F, (, ++ y3)
(12)
Considering the linearized analysis of equations (I), (2), (5), (6), and (7) and incorporating (8) and (9) lead to a generalized transfer function that is applicable for the extending and retracting cases and is given by
[1+F*w] +FXy(l + y)+ S'Xy%z I+s a(y+ r3) + XY +r3) (1 +r3) +r3) (1 +y3)a [(1 1 (I K,a
(13)
S = sCR,,
Considering transfer function (13), it follows that the linearized dynamic response will be the same in both directions provided that the a values are equal. From equation (12) this is possible, provided that
y-1
F=T-
(-J=
J -
l+Y
(15)
1 + y3
It follows from equation (11) that the steady-state velocities in both directions are then equal. Further consideration of the transfer function shows that it is now necessary only to specify the “time constant ratio” X to satisfy the response overshoot requirement for each area ratio y. A guide to this selection is established by considering the double-rod case, y = 1 and F = 0, where the transfer function simplifies to second-order form. The solution for a step response overshoot of 4.6% gives X = 3.92, and the exact value of the nonlinear system is obtained from the digital simulation. Model 1 in the appendix is used, and the solution is shown in Figure 2. Generally, the actual parameter X to be used is lower than that predicted by linearized theory, since the ex-
(14)
act solution gives more damping than the linearized solutions at the same operating conditions. This is particularly true in this case because the exact solution is valid at maximum rod velocity only. A set of transient responses is shown in Figure 3 for area ratios y = 1, 4,2, at optimum load force, and for the extending case. The responses for the retracting case are identical except that the pressure roles are reversed, that is, Pz appears like P, and vice versa. The velocity responses are identical in magnitude. Single-rod
cylinder,
closed-loop
speed control
Figure 4 shows the closed-loop
steady-state
0.5
1
pressures
\
Y =I
,_----_
‘\.,k_ +_______ L<_----
system in which the for both extending and retract-
2
05 -p
i
Figure 3. Some optimized transient responses for y = 1, +, 2, F = (y - 1112 for the open-loop system
1L 1
Figure 2. Optimum withF= (y1)/2
600
Appl.
Math.
1.2
1.4 area
1.6 rat10
1.8
2
Y
X for the open-loop
speed control system Figure 4.
Modelling,
1990,
Vol. 14, November
Closed-loop
speed control of a single-rod cylinder
Optimum
ing cases are identical with the open-loop system previously discussed. The steady-state velocity now depends upon the gain, KU, as well as the load force F. Hence the steady-state servovalve current is influenced by these parameters, and the linearized servovalve resistances are different from R, used in the purely open-loop system. For this reason, transfer function (13) cannot be used for closedloop analysis. It is an easy matter to show that the steady-state current, i,, is given by ~,ef 1,
=
iref = G, ff,i
1 + K,,a’
-=
Ud,
ud=$
U a
K
aud(s)
1 + K,a +-
s
and therefore u -= r/Cl
velocity,
U, is given by
K,a 1 + K,,a
(17)
where a is as defined in equation (12) and is different for the extending and retracting cases. Linearizing the system equations now gives the following closed-loop transfer function:
1
+
s(1 + K,aMy + r3) (1 + r3)
a(1 + K,a)2(y + y3) +
(1 + r3)
the steady-state
(16)
I‘
6Ws)
response design for control systems: J. Watton
Again, as for the open-loop system, the linearized dynamic response is identical in either direction, provided that equation (15) is satisfied. This is also an easy matter to show that for the double-rod case, y = 1 and F = 0, the resulting second-order transfer function satisfies the optimum response criterion provided that X = 3.92(1 + KU)3. This again is used as a design guide when considering the exact simulation results. In this case, Model 2 in the appendix is used, and the requirement of a 4.6% overshoot in response to a step input gives the results shown in Figure 5 for a range of areas ratios, y, and gain, K,,. The results presented in Figure 5 show how the design parameter X must increase as the system gain, K,,, is increased. All the values fall below the linearized solution for y = 1, but of course they converge as K,, is decreased such that the dynamic velocity fluctuation from the initial condition is reduced. Some transient responses are shown in Figure 6 for an area ratio y = 4 and different values of the system gain. It will be observed from Figure 6 that cavitation may occur as the system gain is increased beyond a
I +S2Xy(l +y)+
1
XY a(1 + K,a)
S3Xy2a(l + K,,a)
(1 + r3)
(1 + Y’)
value typically K,, = 1. Similar conclusions y = 1 and y = 2. Single-rod
cylinder,
closed-loop
(18)
hold for
position control
Figure
7 illustrates the closed-loop position control system utilizing a transducer with gain HP. This control system using a symmetrically underlapped spool has been studied for the variable volume case,2’ and the concept of a steady-state position error that varies with load force was introduced. It has been shown that a zero steady-state error, obtained from the exact solution, is only zero provided that
(19)
Figure 6. Some optimized transient responses for y = 4, F = a for the closed-loop velocity control of a cylinder
I I
I llllill
0.1
Figuy_e5. Optimum andF= (y1)/2
i
’ Xfor
Ku
,
Iilllil IO
the closed-loop speed control system Figure 7.
Closed-loop
Appl. Math. Modelling,
position control of a single-rod
1990, Vol. 14, November
cylinder
601
Optimum
response design for control systems: J. Watton
This is remarkably the same requirement as for dynamic response symmetry for the speed control systems previously analyzed. Therefore the generalized closed-loop transfer function, which always indicates
-SYY(S) = @d(S)
,
+
St1
+
r2)
+
2( 1 + Y)K,
a zero position error, is truly valid only for one load force. Since the steady-state pressures are always equal to half the supply pressure, this transfer function simplifies to the following form:
1 S2XY ?XY (1 + Y)K, + 2(1 + Y)&
(20)
where FE
l
y-
2
K
’
P,
GJfpKf
=
A2
’
S = sCR,,,
J 1’
Ep = K&R,,
Yd
=
x = 0.356(1 + Y2)
P (1
+
Y
Y)
=
yd
Yref =
Yref
=m
(21)
k&,
(22)
(23)
Therefore, a particular area ratio is associated with a unique system gain and time constant ratio parameter. The computer simulation of the exact system equations is again used to reassess the linearized analysis prediction, and the proposal is that the parameter X be as defined in (23) and the gain K,, be modified to give the transient response overshoot of 4.6%. This can again be done for a range of area ratios y using Model 3 given in the appendix. However, since position control is perhaps more common than velocity control, the effect of position demand on the optimum performance has also been studied. Figure 8 therefore shows the system gain required for nondimensional demanded positions of yd = I, 2, 4, 8, where the reference position is given by Ld
I,
VdIHp
It is now possible to match the third-order ITA criterion to transfer function (20), and the solution is K = 0.266(1 + Y2)
R
F
The effect of an increased demanded position is to slightly increase the gain required to give the desired optimum response. However, retaining the idealized value given in (23) will result in an adequate response for a wide range of operating conditions. A typical set of transient responses are shown in Figure 9 for an area ratio y = 4 and for a range of demanded positions. The effect of an increased demanded position is to probably induce cavitation as the pressure Pi (when extending) falls to zero. This tendency to cavitate is removed as the area ratio is decreased toward Y = 1, the double-rod case. Conclusions The linearized analysis technique was found to be a useful guide to determining an adequate optimum response for the three configurations considered. It was found in all cases that symmetry of dynamic resnonse mav be achieved by matching the supply pressure to the load. This unique setting for both ve-
(24)
2
9
L
T&=8
B
---..__
,’
2
,_y--__
, __--___1__ IC
0.1
II III1
Ol
5t/t
-__4 -.2 -----_ , ..- l,neor,sed analysts -
I
!
I 1.2
I
I
1
I I,
I
I I I 1.6 I.4 area ratm Y
I
I
i
11
I
I.8
I]
Appl.
Math.
Modelling,
1990,
1
I
I 2
Figure 8. Optimized system gain from closed-loop control and for 7 = (y - 1)/2, X = 0.35611 + y*)
602
10
position
Vol. 14, November
Figure 9. Some optimized transient responses for y = j, F = t and for closed-loop position control of a cylinder
Optimum
locity and position control is achieved by using F = (y - 1)/2 or P, = 2Flrod area. It follows that the steady-state velocities are identical in both directions for the open-loop and closed-loop velocity control systems. For the range of area ratios considered, y = 1-2, the design parameter X is lower than the linearized solution for the open-loop and the closed-loop velocity control system. This follows an anticipated trend, since small-signal linearization inherently suggests lower damping than the large signal practical case.’ Hence a lower X-value, suggesting a lower inertia to give more overshoot, is required to satisfy the response criterion. For the closed-loop velocity control system a working design rule would be X = 3. I( 1 + KIJZ and adequately embraces the area ratios of interest. The optimum ITAE criterion for position control requires specific values of both the-time constant ratio X and the nondimensional gain KP: The design approach adopted is to assume the optimum X required, that is, X = 0.356(1 + r2)lr and then select the gain term from the exact simulation. This gives a solution again always above that suggested by a linearized analysis, the increase depending upon the magnitude of the movement required. The trend corresponds to the linearized solution and for the demand positions studied, &yref = 1-8, a satisfactory design rule would be K, = 0.29(1 + y’)l(l + $. Large pressure fluctuations were found to occur in many instances for each of the three control systems. For the open-loop velocity control system, maximum and minimum pressures were relatively insensitive to the area ratio selected, and in all the cases studied these were close to supply pressure and zero gauge pressure respectively. Similar comments hold true for closedloop velocity control, particularly as the system gain is increased. Closed-loop position control is more tolerant of large pressure fluctuations, but the limits of supply pressure and zero gauge pressure will inevitably be reached for large demanded changes in position. In all cases, therefore, it might be worth considering pressure feedback and reassessing the response criteria.
ud u ref
V vd
X Y Yd
Yref P Y 7
References I 2
3 4
5
F, F G, f& HU i, T lref 1,
kf
-
KP’
K/J
Watton, J. Fluid Power alog and Microcomputer
Systems: Modeling, Simulation, AnControl. Prentice-Hall, Englewood
Cliffs, N.J., 1989 Martin, K. F. Stability and step response of a hydraulic servo with special reference to unsymmetrical oil volume conditions. J. Mech. Engrg. Sci. 1970, U(5), 331-338 Urata, E. Influence of compressibility of oil on the step response of a hydraulic servomechanism-the response under inertia loading. Bull. JSME 1982, 25(203), 797-803 Takahashi, K. and Takahashi, Y. Dynamic characteristics of a spool valve controlled servomotor with a non-symmetrical cylinder. Bull. JSME 1980, 23(10), 1155-l 162 Goodwin, A. B. Fluid Power Svstems. Macmillan. New York, I979
6
Watton, J. The effect of drain orifice damping on the performance characteristics of a servovalve flapper/nozzle stage. J.
7
Zaborszky, J. and Harrington, H. J. Generalised charts of the effects of non-linearities in 2 stage electrohydraulic control valves. AIEE Trans. 1958, 76, 401-408 Feng, T. Y. Static and dynamic control characteristics of flapper-nozzle valves. ASME 1. Basic Engrg. 1959, 81, 275-284 Merritt, H. E. Hydrclulic Control System. John Wiley and Sons, New York, 1967 Nikiforuk, P. N., Ukrainetz, P. R. and Tsai, S. C. Detailed analysis of a two-stage four-way electrohydraulic flow-control valve. J. Mech. Engrg. Sci. 1969, 11(2), 168-174 Martin, D. J. and Burrows, C. R. The dynamic characteristics of an electrohydraulic servovalve. ASMEJ. Dynamic Systems, Measurement and Contra/ Dec. 1976, 395-406 de Pennington, A., ‘t Mannetje, J. J. and Bell, R. The modelling of electrohydraulic control valves and its influence on the design of electrohydraulic drives. J. Mech. Engrg. Sci. 1974,
Dynamic
8 9 IO
linearized parameter cylinder areas fluid capacitance (V/p) cylinder load force and nondimensional value (F/P,AJ servoamplifier gain position transducer gain velocity transducer gain servovalve current and nondimensional value (i/i,,) reference servovalve current servovalve current at the spool underlap boundary servovalve flow constant position gain and nondimensional value (KZ’R)
velocity gain load inductance (m/AIA2) load mass pressures on either side of the system supply pressure to the servovalve flow rates on either side of the actuator servovalve linearized resistances Laplace operator and nondimensional value (SW cylinder velocity demand velocity reference velocity volume on either side of actuator control system demand voltage time constant ratio (L/CR2) cylinder position demanded cylinder position reference cylinder position fluid bulk modulus cylinder area ratio (Al/A,) simulation reference time constant
u
Notation
:I, A* c
response design for control systems: J. Watton
II 12
I3 14 I5 16 17
Appl.
Systems,
Measurement
and Control
1987,109,19-23
16(3), 196-204 Transfer Function for Dowty-Moog
Servovalves. Available from the manufacturer. Watton, J. Fluid power design by CAD. Professional Engrg. March 1989, 33-34 Backe, W. and Hoffmann, W. DSH programme for digital simulation of hydraulic systems. Paper Cl, BHRA 6th International Fluid Power Symposium, Cambridge, UK, 1981 Kinoglu, F. et al. Streamlining hydraulic circuit designs with computer aid. Comput. Mech. Engrg. 1982, l(2), 21-26 Vilenius, M. J. The application of sensitivity analysis to electrohydraulic position servos. J. Dynamic Systems, Measurement and Control 1983, 105, 77-82
Math. Modelling,
1990, Vol. 14, November
603
Optimum 18 19 20
response design for control systems: J. Watton
Karnopp, D. Bond graph models for fluid dynamic systems. J. Dynamic Systems Measurement and Control 1972, Series G, 94(3), 222-229 Thoma, J. U. Simulation by Bondgraphs. Springer-Vet-lag, New York, 1989 Watton, J. The generalised response of servovalve-controlled, single-rod, linear actuators and the influence of transmission line dynamics. .I. Dynamic Systems, Measurement and Control 1984, 106, 157-162
21
22 23
Watton, J. The effect of servovalve underlap on the accuracy and dynamic response of single-rod actuator position control systems. J. Fluid Control 1988, B(3), 7-24 Ogata, K. Modern Control Engineering. Prentice-Hall, New York, 1970 Graham, D. and Lathrop, R. C. The synthesis of optimum response criteria and standard forms. AZEE Trans. Part II 1953,
x=&g 0
Owing to the matched load such that F = FIP,A2 = 1)/2, the retracting case gives identical velocity responses and pressure responses with the roles of P, and P2 interchanged. Single-rod cylinder, closed-loop velocity control (Model 2) extending: (y -
72, 273-288
24
Baz, A. Optimisation of the dynamics of pressure compensated axial piston pumps. J. Fluid Control 1983, 15, 64-81
Appendix: System exact equations
dp,
The appropriate servovalve flow equations (1) and (2) or (3) and (4) are combined with the actuator flow continuity and momentum equations (5), (6), and (7). These equations are then nondimensionalized in a manner appropriate to the system being studied, the reference values being
jY=-__
F
(y-
PsAz
dt/7 = K,( 1 - @V/2(1 - P,) - yu dp, -z dtl7
i? - K,(l
- u)m
1) 2
Single-rod cylinder-open loop velocity control (Model 1) extending:
Again design symmetry results in the retracting case giving similar conclusions as for the open-loop velocity control system. Note that for the closed-loop system the reference time, 7, is a function of the system gain, K,. Single-rod cylinder, closed loop position control (Model 3):
-
$i
= ti
Z,(l + i)m
$T
sign (1 - P,) - Z,( 1 - ;)fi]
[
= ti
Zr(l + t)fl
- Zz(l - i)U/FFJ
sign (1 - P2)
[
- $9
1
+ V
df? dt/r 2 dtly
Yref
Zl?
LLJJ
sCP A2
CR,
I
604
Appl.
Math. Modelling,
z2t
1990,
Vol. 14, November