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State-space Model of an Electro-Hydraulic Actuated Wet Clutch
6th IFAC Symposium Advances in Automotive Control Munich, Germany, July 12-14, 2010
State-space Model of an Electro-Hydraulic Actuated Wet Clutch A. E. Balau and C. Lazar Department of Automatic Control and Applied Informatics ”Gheorghe Asachi” Technical University of Iasi, Str. Prof.Dr.docent Dimitrie Mangeron, nr. 27, 700050, Romania (e-mail:{abalau, clazar}@ac.tuiasi.ro) Abstract: Automotive engineers are continuously exploring various transmission technologies to increase vehicle performance, fuel economy and safety. The most common control devices inside the transmission are hydraulic actuated clutches. In this paper a state-space model for a wet clutch actuated by an electrohydraulic valve used by Volkswagen for automatic transmission is presented. The developed model was implemented in Matlab-Simulink and was validated against the data obtained from a test bench provided by Continental Automotive Romania. The simulations are very similar to the experimental data proving that the modeling approach is suitable to this kind of clutch actuated by an electro-hydraulic pressure reducing valve. Keywords: Wet Clutch, Hydraulic Actuator, Dynamic Modeling constructed based on gray-box approach which combines mathematical modeling and system identification for an electro-magnetic control valve (Tai and Tsao, 2002), inputoutput simplify mathematical model and a state-space mathematical model for an electro-hydraulic valve actuator, both based on system identification and physical laws (Balau et al. 2009).
1. INTRODUCTION During the last few years, the interest for automated manual transmission (AMT) systems has increased due to growing demand of driving comfort. Automated clutch actuation makes it easier for the driver, particularly in stop and go traffic, and has especially seen a recent growth in the European automotive industry. An AMT system consists of a manual transmission through the clutch disc, and an automated actuated clutch during gear shifts. Some of AMT’s largest advantages are low cost, high efficiency, reduced clutch wear and improved fuel consumption.
During the last years models of the multi-plate wet clutches and theirs actuators are reported in literature: (Edelaar, 1997) presents a model for a pressure reducing valve used as actuator for a wet clutch, a dynamic model of electrohydraulic wet clutches is illustrated in (Morselli et al., 2003), a controller for an electro-pneumatic valve used as clutch actuator is developed in (Langjord et al., 2008), a model for an electro-hydraulic valve used as actuator for a wet clutch is presented in (Morselli and Zanasi, 2006).
The basic function of any type of automotive transmission is to transfer the engine torque to the vehicle with the desired ratio smoothly and efficiently. The most common control devices inside the transmission are clutches and hydraulic pistons. Such clutches can be hydraulic actuated, motor driven or actuated using other means. The automatic control of the clutch engagement plays a crucial role in AMT vehicles, and in this paper we deal with the problem of modelling an electro-hydraulic actuated wet clutch.
In (Balau et al. 2009) two models for an electro-hydraulic actuator were developed: an input-output model, where simplifications were made in order to obtain a suitable transfer function to be implemented in Matlab-Simulink, and a state-space model. Starting from the actuator state-space model, in this paper a state-space model for a wet clutch actuated by an electro-hydraulic valve used by Volkswagen for automatic transmission is presented. No simplifications were made in order to develop the model and the simulations show that the state-space model captures the essential dynamics of the system. The electro-hydraulic actuated clutch model has as input the electric current, the line pressure and the tank pressure, and as outputs the reduced pressure, the clutch pressure, the clutch piston displacement and the valve plunger displacement. The developed model was implemented in Matlab-Simulink and the results obtained with the developed simulator were validated against the data obtained from a test bench provided by Continental
Recent attention has focused on modeling and developing advanced control methods for different valve types used as actuators in automotive control systems: physics-based nonlinear model for an exhausting valve (J. Ma et al., 2008), nonlinear state-space model description of the actuator that is derived based on physical principles and parameter identification (Y. Wang et al., 2002), (Peterson et al., 2003), nonlinear physical model for programmable valves (Liu and Yao, 2008), nonlinear model of an electromagnetic actuator used in brake system based on system identification (A. Forrai, et al., 2005), mathematical model obtained using identification methods for a valve actuation system of an electro-hydraulic engine (Liao et al., 2008), linear model 978-3-902661-72-2/10/$20.00 © 2010 IFAC
506
10.3182/20100712-3-DE-2013.00039
AAC 2010 Munich, Germany, July 12-14, 2010
Automotive Romania which includes the Volkswagen wet clutch actuated by the electro-hydraulic valve.
by a voltage u, applied to a solenoid, and it is a nonlinear function of the electric current i and the displacement x :
The rest of the paper is organized as follows. In Section 2, the structure and the functional operation of the electro-hydraulic actuated wet clutch are presented and a state-space mathematical model of the valve-clutch system is developed. Section 3 presents the block diagram and the parameters used in simulations. Also, simulation results for the Simulink model are discussed. The concluding remarks are given in Section 4.
Fmag = f (i, x ) =
ka i 2 2 ( kb + x )
2
; L
di + Ri = u , dt
where ka and kb are constants, L is the solenoid induction and R the resistance. The charging phase of the pressure reducing valve has been illustrated in Fig.1. A positive displacement of the spool allows connection between the source and the hydraulic clutch, while the channel that connects the hydraulic clutch with the tank is kept closed.
2. STATE-SPACE MODEL OF AN ACTUATOR AND CLUTCH SYSTEM The model designed in this Section for an electro-hydraulic actuated wet clutch is based on physical principles for flow and fluid dynamics.
Line pressure
2.1 Structure and functional operation
PS
x>0
Valve plunger D, PD
Schematics of the electro-hydraulic valve actuator and wet multi-plate clutch layout are given in Fig. 1. A pump produces the line pressure PS used as input for the electrohydraulic actuator represented by a pressure reducing valve. This valve realises a pressure PR on the clutch side, depending on the current i in the solenoid, which will have as consequence the magnetic force Fmag exerted on the valve plunger. The pressure to be controlled PR is sensed on the plunger end areas C and D and compared with the magnetic force Fmag . The two forces FC and FD from the two sensed pressure chambers generate the feedback force Ffeed = FC − FD . Depending on the valve plunger position,
(1)
A,, P C PCC K1
FC
FD
KC QC
QD
Solenoid Fmag
K2
i
u
PR
QL
Input cooling oil
K3
PL , AL
AL
PT
Tank
PL Output axle
Input axle Reset spring
Piston stop
Output cooling oil to tank
there are two phases: the charging phase, when the magnetic force is grater than the feedback force and the valve plunger is moved to the left, connecting the source with the hydraulic actuated clutch (Fig. 1), and the discharging phase, when the magnetic force is switched off or has a lower value than the feedback force so that the valve plunger is moved to the right, connecting the hydraulic actuated clutch to the tank (Fig. 2).
Fig. 1. Actuator and clutch system: Charging phase Using the magnetic force and the feedback force it results a force balance which describes the plunger motion. This equation of force balance is the same for both positive and negative displacement of the plunger:
The wet clutch is a chamber with a piston as represented in (Fig. 1). In the charging phase when the valve plunger is moved to the left and the displacement x is considered positive, the oil flows through the valve to the clutch and the piston in the clutch moves towards the clutch plates compressing them. In the discharging phase, when the valve plunger is moved to the right and the displacement x is negative, the clutch piston moves to the left and the oil flows from the clutch chamber through the valve to the tank.
Mv
dv = Fmag − CPC + DPD − K e x(t ) , dt
(2)
where C , PC represent the area and the pressure of the left sensed chamber, D, PD the area and the pressure of the right sensed chamber, M v is the plunger mass,
dv represents the dt
2.2 State-space mathematical model of the actuator-clutch system
acceleration and v(t ) =
The valve plunger movement is due to the difference between the feedback force Ffeed and the magnetic force Fmag , the
PS is the supply pressure, PR is the reduced pressure,
dx the velocity of the plunger, dt K e = 0.43w( PS0 − PR0 ) represents the flow force spring rate,
charging and the discharging phases being illustrated by the sign of the resulting force. The magnetic force is generated
PS0 , PR0 are the nominal values of the pressures and w represents the area gradient of main orifice. 507
AAC 2010 Munich, Germany, July 12-14, 2010
The linearized continuity equations which describe the dynamics from the sensed pressure chambers are:
Vc d P (t ) = − K1 ( PC (t ) − PR (t )) + Cv(t ) , β e dt C
(7) (8) (9)
Vc d PC (t ) = K1 ( PR (t ) − PC (t )) + Cv (t ) , β e dt
(3)
VD d PD (t ) = − K 2 ( PD (t ) − PR (t )) − Dv(t ) , β e dt
VD d PD ( t ) = K 2 ( PR ( t ) − PD ( t )) − D v ( t ) , β e dt
(4)
VL d PL (t ) = − K 3 ( PL (t ) − PR (t )) − AL v(t ) . β e dt
Using the flow through the left and right sensed chambers, the flow through the main orifice (from the hydraulic clutch to the tank) and the clutch flow, the linearized continuity equation obtained for the chamber of the pressure being controlled is:
where K1 , K 2 are the flow-pressure coefficients of restrictors,
VC ,VD are the sensing chamber volumes, PC , PD are the pressures from the two chambers and β e represents the effective bulk modulus. Applying the continuity equation to the clutch piston chamber yields:
VL d P (t ) = K3 ( PR (t ) − PL (t )) − AL v(t ) , βe dt L
Vt d PR (t ) = K q x (t ) − K1 ( PR (t ) − PC (t )) − K ce PR (t ) − β e dt ,(10) − K 2 ( PR (t ) − PD (t )) − K 3 ( PR (t ) − PL (t )) + K D PT (t )
(5) where K D is the flow-pressure coefficient of main orifice and PT represents the tank pressure.
where K3 is the flow-pressure coefficient of pipe from valve actuator to the clutch, VL the piston chamber volume, PL the
The valve dynamics in the discharging phase of the electrohydraulic actuated clutch is illustrated by the equations (2), (7), (8), (9) and (10).
pressure in the clutch chamber and AL the area of the clutch piston. Using the flow through the left and right sensed chambers, the flow through the main orifice (from the source to the hydraulic clutch) and the flow through the clutch chamber, the linearized continuity equation at the chamber of the pressure being controlled is:
Line pressure
FC
A, CP , CPC K1
FD
PR QL
Input cooling oil
where KC is the flow-pressure coefficient of main orifice,
K ce = K C + Kl
Reset spring
leakage coefficient and Vt represents the total volume of the chamber where the pressure is being controlled.
Piston stop
QD
K2
u
QT
K3
PT
PL, AL
AL
Tank
PL
Input axle
represents the equivalent flow-pressure coefficient, Kl is the
Solenoid
Fmag
i
KD
QC
+ K 2 ( PD (t ) − PR (t )) + K 3 ( PL (t ) − PR (t )) + K C PS (t )
is the flow gain of main orifice,
x <0
D, PD
Vt d P (t ) = K q x(t ) + K1 ( PC (t ) − PR (t )) − K ce PR (t ) + β e dt R ,(6)
Kq
PS
Output axle
Output cooling oil to tank
The equations (2), (3), (4), (5) and (6) define the valve dynamics in the charging phase for the electro-hydraulic valve actuated clutch. A negative displacement of the pressure reducing valve spool allows connection between the hydraulic actuated clutch and the tank, while the channel that connects the source with the hydraulic clutch is kept closed.
Fig. 2. Actuator and clutch system: Discharging phase To model the clutch piston dynamics, Newton’s second law is applied to the forces that actuates on the clutch piston, resulting:
The linearized continuity equations at the sensed pressure chambers and the clutch chamber for the discharging phase of the valve, illustrated in Fig.2, are:
Mt
508
dv p dt
= AL PL (t ) − B f v p (t ) − Kx p (t ) ,
(11)
AAC 2010 Munich, Germany, July 12-14, 2010
where x p the piston displacement, M p total mass of the
while different values of the B matrix are used: B1 for the
piston, K is the load spring gradient, B f is the viscous
charging phase of the system and B 2 for the discharging phase of the electro-hydraulic actuated clutch:
damping coefficient of the piston, acceleration and v p (t ) =
dx p
dv p dt
represents the
the velocity of the clutch
dt
piston. Combining the equations (2), (3), (4), (5), (6), (11) that describe the dynamics of the system in the charging phase, and equations (2), (7), (8), (9), (10) and (11) that describe the dynamics of the system in the discharging phase of the system, the state-space model of the electro-hydraulic actuated clutch is design according to:
x& ( t ) = Ax ( t ) + Bu ( t ) . y ( t ) = Cx ( t ) + Du ( t )
B1
(12)
0 0 0 0 = 0 0 Kcβe V t 0
0 0 0 0 0 0 0 0
1 0 Mv 0 0 0 0 0 0 ,B2 = 0 0 0 0 0 0 0 0
0 0 0 0 0 0
Kcβe Vt 0
1 Mv 0 0 0 . 0 0 0 0
(15)
3. SIMULATIONS AND EXPERIMENTAL RESULTS Starting from the equations that illustrates the mathematical model, a block diagram for the actuator-clutch system was created (Fig. 3). It can be seen that a switch block was used, relative to the sign of the plunger displacement, in order to commutate between the two phases that describe the functionality of the actuator, B1 for the charging phase of the
The state vector is represented by x ( t ) , u ( t ) is the input vector, and y ( t ) is considered to be the output of the system:
system and B 2 for the discharging phase.
T
u ( t ) = PS ( t ) PT ( t ) Fmag ( t ) , T
x ( t ) = v ( t ) x ( t ) vp ( t ) xp ( t ) PC ( t ) PD ( t ) PR ( t ) PL ( t ) . (13) T
y(t ) = x ( t ) xp ( t ) PR ( t ) PL ( t ) .
Instead of the solenoid current, the magnetic force was used as input because it is a nonlinear function of the current, the relation between the magnetic force and the current together with the plunger displacement being implemented in a form of a two dimensional look-up table designed at Continental Automotive Romania, for this type of valve. The matrix A is the same for both charging and discharging phase of the actuator-clutch system and we consider K sum = K ce + K1 + K 2 + K3 in: K − e 0 M v 1 0 0 0 0 0 Cβe 0 A= Vc − Dβe 0 V D Kqβe 0 Vt 0 0
Fig. 3. Block diagram of actuator-clutch system In order to validate the model obtained for the electrohydraulic actuated clutch, a simulator was designed and developed in Matlab/Simulink starting from the mathematical model described in Section 2. The parameters used in the model of the valve actuated wet clutch, summarized in Appendix A, are estimated experimentally at Continental Automotive Romania using a test-bench, or are already given by the manufacturer.
0 0 0 0 0 0 AL K 0 0 0 0 Mt Mt 1 0 0 0 0 0 Kβ K1βe 0 0 − 1e 0 0 , (14) Vc Vc Kβ K2βe 0 0 0 − 2 e 0 VD VD K1βe K2βe Ksumβe K3βe 0 0 − Vt Vt Vt Vt ALβe K3βe Kβ 0 0 0 − 3 e VL VL VL
0
0
−
C Mv
D Mv
0
0
A test-bench which includes the Volkswagen DQ250 wet clutch actuated by the electro-hydraulic valve DQ500 was provided by Continental Automotive Romania. Experiments made on the test-bench allowed obtaining the parameters used in simulations for the electro-hydraulic actuator: the volumes of the actuator chambers, the left and right areas of the valve plunger, the flow-pressure coefficients. The testbench also provides measurements for the outputs of the system, represented by the valve plunger displacement and the clutch pressure which are used in order to validate the implemented simulator.
509
AAC 2010 Munich, Germany, July 12-14, 2010
Following experiments made on the test-bench from Continental Automotive Romania, using as input the solenoid current i, respectively the magnetic force Fmag obtained
the pressure from the source, the tank pressure and the clutch flow. The clutch flow was simulated in a form of two impulse signal, one for the charging phase and one for the discharging phase of the actuator. The state-space model is more precise, because no simplifications were made in order to describe the system, as for the input-output model.
through the look-up table, and represented in Fig. 4, the realtime clutch pressure response PL from Fig. 5 was obtained.
0.8
Fmag Current (i)
4
0.6 Punger displacement [mm]
Current [A], Fmag [N]
5
3 2 1 0 0.5
0.4 0.2 0 -0.2 -0.4 -0.6
1
1.5
2 Time [s]
2.5
3
3.5 -0.8 0.5
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 4. Input signals Fig. 6. Valve plunger displacement The state-space model simulation results are validated due to similar behaviour obtained for the pressure in the clutch chamber. Using as input signal the same electric current i from the experiments made on the test-bench, Fig. 5 shows the comparison between measurements and simulations for the pressure obtained in the clutch chamber, along with the simulation results of the reduced pressure. Good agreement between the real-time and simulation results proves that the model captures the essential dynamics of the system.
In the state-space model developed in this paper for the actuator-clutch system, the clutch flow, which is illustrated in Fig. 7, was obtained as a difference between the pressure from the valve and the clutch pressure. -3
3 2
Clutch Flow [m 3/s]
6
5
4 Pressure[bar]
x 10
1 0 -1 -2
3
-3
2
-4 0.5
1
0 0.5
simulated PR simulated PL measured PL 1
1.5
2 Time[s]
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 7. Clutch flow 2.5
3
3.5
The simulation results obtained for the clutch piston displacements are illustrated in Fig. 8, results obtained with the same input signal from Fig. 4.
Fig. 5. Pressures behaviour It can be seen that for a positive clutch flow, there are positive displacements both for the valve plunger and the clutch piston, illustrating the charging phase of the valve when the clutch chamber is filled with oil, while for a negative clutch flow, there are negative displacements, illustrating the discharging phase of the valve and the oil going from the clutch chamber through the valve to the tank.
The behaviour obtained for the valve displacement represented in Fig. 6, is similar with the behaviour obtained in (Balau et al. 2009) where a linearized input-output model and a state-space model were developed for the electrohydraulic pressure reducing valve used as actuator for the wet clutch. The inputs were represented by the magnetic force, 510
AAC 2010 Munich, Germany, July 12-14, 2010
using on/off valves, In: Proc. of 2008 American Control Conference, Seattle, USA, June 11-13. Liao H.-H., M.J. Roelle, and J.C. Gerdes (2008). Repetitive control of an electro-hydraulic engine valve actuation system, In: Proc. of American Control Conference, Seattle, Washington, USA, pp. 957-980. Liu S. and B. Yao (2008). Coordinative control of energy saving programmable valves, In: IEEE Transactions on Control Systems Technology, 16, pp. 34-45. Ma J., G. Zhu, A. Hartsig, and H. Schock (2008). Modelbased predictive control of an electro-pneumatic exhaust valve for internal combustion engines, In: Proc. of American Control Conference, Seattle, Washington, 1113 June, USA, pp. 298-305. Morselli R. and R. Zanasi (2006). Modeling of Automotive Control Systems Using Power Oriented Graphs, In: Proc. of 32nd Annual Conference on Industrial Electronics, IECO? 2006, Paris, France, pp. 5295 – 5300. Peterson K.S., A.G. Stefanopoulou, Y. Wang, and T. Megli (2003). Virtual lash adjuster for an electromechanical valve actuator through iterative learning control, In: Proc. of International Mechanical Engineering Congress, Washington, 15-21 Nov., USA. Tai C. and T.-C. Tsao (2002). Control of an electromechanical camless valve actuator, In: Proc. of American Control Conference, Anchorage, 8-10 May, USA, pp. 262-267. Wang Y., T. Megli, and M. Haghgooie (2002). Modeling and control of electromechanical valve actuator, In: Society of Automotive Engineers, pp. 1-10.
450
Piston displacement [mm]
400 350 300 250 200 150 100 50 0 0.5
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 8. Clutch piston displacement The value obtained for the valve piston displacement is in the range of [-1,+1] mm, like it is supposed to be, because the actuator is a closed loop system and the plunger displacement is restricted by the balance in forces. On the other hand, the clutch is an open loop system, with no feedback, resulting a high value of the piston displacement which can be further limited by designing a proper controller for the electrohydraulic actuated wet clutch. 4. CONCLUSION In this paper a state-space model for an electro-hydraulic actuated clutch used in the automotive control systems for automatic transmission was developed. The model was validated by comparing the results with data obtained on a real test-bench provided by Continental Automotive Romania, which includes a Volkswagen wet clutch actuated by an electro-hydraulic valve. It can be concluded that the simulator has good results illustrated by the similar behaviour obtained for clutch pressure compared with the real measured values.
APPENDIX A. COEFFICIENTS VALUES
ACKNOWLEDGMENT The work was supported by National Center for Programs Management from Romania under the research grant SICONA - 12100/2008. REFERENCES Balau A. E., C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H. Matcovschi and O. Pastravanu (2009). Modeling of a Pressure Reducing Valve Actuator for Automotive Applications, In: Proc. of 18th IEEE International Conference on Control Applications, July 8-10, Saint Petersburg, Russia, pp. 1356-1361. Edelaar M.J.W.H. (1997). Model and control of a wet plate clutch, Ph.D Thesis, Eindhoven. Forrai A., T. Ueda, and T. Yumura (2005). A simple approach to electromagnetic actuator control based on asymptotically exact linearization, In: Archive of Applied Mechanics, 74(8), pp. 550-562. Langjord H., T. A. Johansen and J. P. Hespanha (2008). Switched control of an electropneumatic clutch actuator 511
Symbol
Value/Unit
Symbol
Value/Unit
Ke
1000[N/m]
PT
0[N/m^2]
Mv
25e-3[kg]
w
3e-3[m]
βe
1.6e+9[N/m^2]
VC
7.53e-8[m^3]
ω1 ω2 ω3
1.17e+6[rad/s]
VD
1.04e-7[m^3]
5.42e+7[rad/s]
Vt
3.2e-4[m^3]
8.04e+5[rad/s]
C
3.66e-5[m^2]
ω5
1.04e+6[rad/s]
D
2.94e-5[m^2]
KC =KD
7.58e-11[(m^3/s)/(N/m^2)]
α
2e-5
K1
5.50e-10[(m^3/s)/(N/m^2)]
Mp
0.5
K2
3.52e-9[(m^3/s)/(N/m^2)]
K3
1.26e-8
Kq
5.3418[(m^3/s)/m]
VL
2.51e-5
Kce
2.08e-9[(m^3/s)/(N/m^2)]
Al
7.75e-4 900
Kl
2e-9[(m^3/s)/(N/m^2)]
K
Ps
1e+6[N/m^2]
Bf
0
Ka
0.005
Kb
0.01
State-space Model of an Electro-Hydraulic Actuated Wet Clutch A. E. Balau and C. Lazar Department of Automatic Control and Applied Informatics ”Gheorghe Asachi” Technical University of Iasi, Str. Prof.Dr.docent Dimitrie Mangeron, nr. 27, 700050, Romania (e-mail:{abalau, clazar}@ac.tuiasi.ro) Abstract: Automotive engineers are continuously exploring various transmission technologies to increase vehicle performance, fuel economy and safety. The most common control devices inside the transmission are hydraulic actuated clutches. In this paper a state-space model for a wet clutch actuated by an electrohydraulic valve used by Volkswagen for automatic transmission is presented. The developed model was implemented in Matlab-Simulink and was validated against the data obtained from a test bench provided by Continental Automotive Romania. The simulations are very similar to the experimental data proving that the modeling approach is suitable to this kind of clutch actuated by an electro-hydraulic pressure reducing valve. Keywords: Wet Clutch, Hydraulic Actuator, Dynamic Modeling constructed based on gray-box approach which combines mathematical modeling and system identification for an electro-magnetic control valve (Tai and Tsao, 2002), inputoutput simplify mathematical model and a state-space mathematical model for an electro-hydraulic valve actuator, both based on system identification and physical laws (Balau et al. 2009).
1. INTRODUCTION During the last few years, the interest for automated manual transmission (AMT) systems has increased due to growing demand of driving comfort. Automated clutch actuation makes it easier for the driver, particularly in stop and go traffic, and has especially seen a recent growth in the European automotive industry. An AMT system consists of a manual transmission through the clutch disc, and an automated actuated clutch during gear shifts. Some of AMT’s largest advantages are low cost, high efficiency, reduced clutch wear and improved fuel consumption.
During the last years models of the multi-plate wet clutches and theirs actuators are reported in literature: (Edelaar, 1997) presents a model for a pressure reducing valve used as actuator for a wet clutch, a dynamic model of electrohydraulic wet clutches is illustrated in (Morselli et al., 2003), a controller for an electro-pneumatic valve used as clutch actuator is developed in (Langjord et al., 2008), a model for an electro-hydraulic valve used as actuator for a wet clutch is presented in (Morselli and Zanasi, 2006).
The basic function of any type of automotive transmission is to transfer the engine torque to the vehicle with the desired ratio smoothly and efficiently. The most common control devices inside the transmission are clutches and hydraulic pistons. Such clutches can be hydraulic actuated, motor driven or actuated using other means. The automatic control of the clutch engagement plays a crucial role in AMT vehicles, and in this paper we deal with the problem of modelling an electro-hydraulic actuated wet clutch.
In (Balau et al. 2009) two models for an electro-hydraulic actuator were developed: an input-output model, where simplifications were made in order to obtain a suitable transfer function to be implemented in Matlab-Simulink, and a state-space model. Starting from the actuator state-space model, in this paper a state-space model for a wet clutch actuated by an electro-hydraulic valve used by Volkswagen for automatic transmission is presented. No simplifications were made in order to develop the model and the simulations show that the state-space model captures the essential dynamics of the system. The electro-hydraulic actuated clutch model has as input the electric current, the line pressure and the tank pressure, and as outputs the reduced pressure, the clutch pressure, the clutch piston displacement and the valve plunger displacement. The developed model was implemented in Matlab-Simulink and the results obtained with the developed simulator were validated against the data obtained from a test bench provided by Continental
Recent attention has focused on modeling and developing advanced control methods for different valve types used as actuators in automotive control systems: physics-based nonlinear model for an exhausting valve (J. Ma et al., 2008), nonlinear state-space model description of the actuator that is derived based on physical principles and parameter identification (Y. Wang et al., 2002), (Peterson et al., 2003), nonlinear physical model for programmable valves (Liu and Yao, 2008), nonlinear model of an electromagnetic actuator used in brake system based on system identification (A. Forrai, et al., 2005), mathematical model obtained using identification methods for a valve actuation system of an electro-hydraulic engine (Liao et al., 2008), linear model 978-3-902661-72-2/10/$20.00 © 2010 IFAC
506
10.3182/20100712-3-DE-2013.00039
AAC 2010 Munich, Germany, July 12-14, 2010
Automotive Romania which includes the Volkswagen wet clutch actuated by the electro-hydraulic valve.
by a voltage u, applied to a solenoid, and it is a nonlinear function of the electric current i and the displacement x :
The rest of the paper is organized as follows. In Section 2, the structure and the functional operation of the electro-hydraulic actuated wet clutch are presented and a state-space mathematical model of the valve-clutch system is developed. Section 3 presents the block diagram and the parameters used in simulations. Also, simulation results for the Simulink model are discussed. The concluding remarks are given in Section 4.
Fmag = f (i, x ) =
ka i 2 2 ( kb + x )
2
; L
di + Ri = u , dt
where ka and kb are constants, L is the solenoid induction and R the resistance. The charging phase of the pressure reducing valve has been illustrated in Fig.1. A positive displacement of the spool allows connection between the source and the hydraulic clutch, while the channel that connects the hydraulic clutch with the tank is kept closed.
2. STATE-SPACE MODEL OF AN ACTUATOR AND CLUTCH SYSTEM The model designed in this Section for an electro-hydraulic actuated wet clutch is based on physical principles for flow and fluid dynamics.
Line pressure
2.1 Structure and functional operation
PS
x>0
Valve plunger D, PD
Schematics of the electro-hydraulic valve actuator and wet multi-plate clutch layout are given in Fig. 1. A pump produces the line pressure PS used as input for the electrohydraulic actuator represented by a pressure reducing valve. This valve realises a pressure PR on the clutch side, depending on the current i in the solenoid, which will have as consequence the magnetic force Fmag exerted on the valve plunger. The pressure to be controlled PR is sensed on the plunger end areas C and D and compared with the magnetic force Fmag . The two forces FC and FD from the two sensed pressure chambers generate the feedback force Ffeed = FC − FD . Depending on the valve plunger position,
(1)
A,, P C PCC K1
FC
FD
KC QC
QD
Solenoid Fmag
K2
i
u
PR
QL
Input cooling oil
K3
PL , AL
AL
PT
Tank
PL Output axle
Input axle Reset spring
Piston stop
Output cooling oil to tank
there are two phases: the charging phase, when the magnetic force is grater than the feedback force and the valve plunger is moved to the left, connecting the source with the hydraulic actuated clutch (Fig. 1), and the discharging phase, when the magnetic force is switched off or has a lower value than the feedback force so that the valve plunger is moved to the right, connecting the hydraulic actuated clutch to the tank (Fig. 2).
Fig. 1. Actuator and clutch system: Charging phase Using the magnetic force and the feedback force it results a force balance which describes the plunger motion. This equation of force balance is the same for both positive and negative displacement of the plunger:
The wet clutch is a chamber with a piston as represented in (Fig. 1). In the charging phase when the valve plunger is moved to the left and the displacement x is considered positive, the oil flows through the valve to the clutch and the piston in the clutch moves towards the clutch plates compressing them. In the discharging phase, when the valve plunger is moved to the right and the displacement x is negative, the clutch piston moves to the left and the oil flows from the clutch chamber through the valve to the tank.
Mv
dv = Fmag − CPC + DPD − K e x(t ) , dt
(2)
where C , PC represent the area and the pressure of the left sensed chamber, D, PD the area and the pressure of the right sensed chamber, M v is the plunger mass,
dv represents the dt
2.2 State-space mathematical model of the actuator-clutch system
acceleration and v(t ) =
The valve plunger movement is due to the difference between the feedback force Ffeed and the magnetic force Fmag , the
PS is the supply pressure, PR is the reduced pressure,
dx the velocity of the plunger, dt K e = 0.43w( PS0 − PR0 ) represents the flow force spring rate,
charging and the discharging phases being illustrated by the sign of the resulting force. The magnetic force is generated
PS0 , PR0 are the nominal values of the pressures and w represents the area gradient of main orifice. 507
AAC 2010 Munich, Germany, July 12-14, 2010
The linearized continuity equations which describe the dynamics from the sensed pressure chambers are:
Vc d P (t ) = − K1 ( PC (t ) − PR (t )) + Cv(t ) , β e dt C
(7) (8) (9)
Vc d PC (t ) = K1 ( PR (t ) − PC (t )) + Cv (t ) , β e dt
(3)
VD d PD (t ) = − K 2 ( PD (t ) − PR (t )) − Dv(t ) , β e dt
VD d PD ( t ) = K 2 ( PR ( t ) − PD ( t )) − D v ( t ) , β e dt
(4)
VL d PL (t ) = − K 3 ( PL (t ) − PR (t )) − AL v(t ) . β e dt
Using the flow through the left and right sensed chambers, the flow through the main orifice (from the hydraulic clutch to the tank) and the clutch flow, the linearized continuity equation obtained for the chamber of the pressure being controlled is:
where K1 , K 2 are the flow-pressure coefficients of restrictors,
VC ,VD are the sensing chamber volumes, PC , PD are the pressures from the two chambers and β e represents the effective bulk modulus. Applying the continuity equation to the clutch piston chamber yields:
VL d P (t ) = K3 ( PR (t ) − PL (t )) − AL v(t ) , βe dt L
Vt d PR (t ) = K q x (t ) − K1 ( PR (t ) − PC (t )) − K ce PR (t ) − β e dt ,(10) − K 2 ( PR (t ) − PD (t )) − K 3 ( PR (t ) − PL (t )) + K D PT (t )
(5) where K D is the flow-pressure coefficient of main orifice and PT represents the tank pressure.
where K3 is the flow-pressure coefficient of pipe from valve actuator to the clutch, VL the piston chamber volume, PL the
The valve dynamics in the discharging phase of the electrohydraulic actuated clutch is illustrated by the equations (2), (7), (8), (9) and (10).
pressure in the clutch chamber and AL the area of the clutch piston. Using the flow through the left and right sensed chambers, the flow through the main orifice (from the source to the hydraulic clutch) and the flow through the clutch chamber, the linearized continuity equation at the chamber of the pressure being controlled is:
Line pressure
FC
A, CP , CPC K1
FD
PR QL
Input cooling oil
where KC is the flow-pressure coefficient of main orifice,
K ce = K C + Kl
Reset spring
leakage coefficient and Vt represents the total volume of the chamber where the pressure is being controlled.
Piston stop
QD
K2
u
QT
K3
PT
PL, AL
AL
Tank
PL
Input axle
represents the equivalent flow-pressure coefficient, Kl is the
Solenoid
Fmag
i
KD
QC
+ K 2 ( PD (t ) − PR (t )) + K 3 ( PL (t ) − PR (t )) + K C PS (t )
is the flow gain of main orifice,
x <0
D, PD
Vt d P (t ) = K q x(t ) + K1 ( PC (t ) − PR (t )) − K ce PR (t ) + β e dt R ,(6)
Kq
PS
Output axle
Output cooling oil to tank
The equations (2), (3), (4), (5) and (6) define the valve dynamics in the charging phase for the electro-hydraulic valve actuated clutch. A negative displacement of the pressure reducing valve spool allows connection between the hydraulic actuated clutch and the tank, while the channel that connects the source with the hydraulic clutch is kept closed.
Fig. 2. Actuator and clutch system: Discharging phase To model the clutch piston dynamics, Newton’s second law is applied to the forces that actuates on the clutch piston, resulting:
The linearized continuity equations at the sensed pressure chambers and the clutch chamber for the discharging phase of the valve, illustrated in Fig.2, are:
Mt
508
dv p dt
= AL PL (t ) − B f v p (t ) − Kx p (t ) ,
(11)
AAC 2010 Munich, Germany, July 12-14, 2010
where x p the piston displacement, M p total mass of the
while different values of the B matrix are used: B1 for the
piston, K is the load spring gradient, B f is the viscous
charging phase of the system and B 2 for the discharging phase of the electro-hydraulic actuated clutch:
damping coefficient of the piston, acceleration and v p (t ) =
dx p
dv p dt
represents the
the velocity of the clutch
dt
piston. Combining the equations (2), (3), (4), (5), (6), (11) that describe the dynamics of the system in the charging phase, and equations (2), (7), (8), (9), (10) and (11) that describe the dynamics of the system in the discharging phase of the system, the state-space model of the electro-hydraulic actuated clutch is design according to:
x& ( t ) = Ax ( t ) + Bu ( t ) . y ( t ) = Cx ( t ) + Du ( t )
B1
(12)
0 0 0 0 = 0 0 Kcβe V t 0
0 0 0 0 0 0 0 0
1 0 Mv 0 0 0 0 0 0 ,B2 = 0 0 0 0 0 0 0 0
0 0 0 0 0 0
Kcβe Vt 0
1 Mv 0 0 0 . 0 0 0 0
(15)
3. SIMULATIONS AND EXPERIMENTAL RESULTS Starting from the equations that illustrates the mathematical model, a block diagram for the actuator-clutch system was created (Fig. 3). It can be seen that a switch block was used, relative to the sign of the plunger displacement, in order to commutate between the two phases that describe the functionality of the actuator, B1 for the charging phase of the
The state vector is represented by x ( t ) , u ( t ) is the input vector, and y ( t ) is considered to be the output of the system:
system and B 2 for the discharging phase.
T
u ( t ) = PS ( t ) PT ( t ) Fmag ( t ) , T
x ( t ) = v ( t ) x ( t ) vp ( t ) xp ( t ) PC ( t ) PD ( t ) PR ( t ) PL ( t ) . (13) T
y(t ) = x ( t ) xp ( t ) PR ( t ) PL ( t ) .
Instead of the solenoid current, the magnetic force was used as input because it is a nonlinear function of the current, the relation between the magnetic force and the current together with the plunger displacement being implemented in a form of a two dimensional look-up table designed at Continental Automotive Romania, for this type of valve. The matrix A is the same for both charging and discharging phase of the actuator-clutch system and we consider K sum = K ce + K1 + K 2 + K3 in: K − e 0 M v 1 0 0 0 0 0 Cβe 0 A= Vc − Dβe 0 V D Kqβe 0 Vt 0 0
Fig. 3. Block diagram of actuator-clutch system In order to validate the model obtained for the electrohydraulic actuated clutch, a simulator was designed and developed in Matlab/Simulink starting from the mathematical model described in Section 2. The parameters used in the model of the valve actuated wet clutch, summarized in Appendix A, are estimated experimentally at Continental Automotive Romania using a test-bench, or are already given by the manufacturer.
0 0 0 0 0 0 AL K 0 0 0 0 Mt Mt 1 0 0 0 0 0 Kβ K1βe 0 0 − 1e 0 0 , (14) Vc Vc Kβ K2βe 0 0 0 − 2 e 0 VD VD K1βe K2βe Ksumβe K3βe 0 0 − Vt Vt Vt Vt ALβe K3βe Kβ 0 0 0 − 3 e VL VL VL
0
0
−
C Mv
D Mv
0
0
A test-bench which includes the Volkswagen DQ250 wet clutch actuated by the electro-hydraulic valve DQ500 was provided by Continental Automotive Romania. Experiments made on the test-bench allowed obtaining the parameters used in simulations for the electro-hydraulic actuator: the volumes of the actuator chambers, the left and right areas of the valve plunger, the flow-pressure coefficients. The testbench also provides measurements for the outputs of the system, represented by the valve plunger displacement and the clutch pressure which are used in order to validate the implemented simulator.
509
AAC 2010 Munich, Germany, July 12-14, 2010
Following experiments made on the test-bench from Continental Automotive Romania, using as input the solenoid current i, respectively the magnetic force Fmag obtained
the pressure from the source, the tank pressure and the clutch flow. The clutch flow was simulated in a form of two impulse signal, one for the charging phase and one for the discharging phase of the actuator. The state-space model is more precise, because no simplifications were made in order to describe the system, as for the input-output model.
through the look-up table, and represented in Fig. 4, the realtime clutch pressure response PL from Fig. 5 was obtained.
0.8
Fmag Current (i)
4
0.6 Punger displacement [mm]
Current [A], Fmag [N]
5
3 2 1 0 0.5
0.4 0.2 0 -0.2 -0.4 -0.6
1
1.5
2 Time [s]
2.5
3
3.5 -0.8 0.5
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 4. Input signals Fig. 6. Valve plunger displacement The state-space model simulation results are validated due to similar behaviour obtained for the pressure in the clutch chamber. Using as input signal the same electric current i from the experiments made on the test-bench, Fig. 5 shows the comparison between measurements and simulations for the pressure obtained in the clutch chamber, along with the simulation results of the reduced pressure. Good agreement between the real-time and simulation results proves that the model captures the essential dynamics of the system.
In the state-space model developed in this paper for the actuator-clutch system, the clutch flow, which is illustrated in Fig. 7, was obtained as a difference between the pressure from the valve and the clutch pressure. -3
3 2
Clutch Flow [m 3/s]
6
5
4 Pressure[bar]
x 10
1 0 -1 -2
3
-3
2
-4 0.5
1
0 0.5
simulated PR simulated PL measured PL 1
1.5
2 Time[s]
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 7. Clutch flow 2.5
3
3.5
The simulation results obtained for the clutch piston displacements are illustrated in Fig. 8, results obtained with the same input signal from Fig. 4.
Fig. 5. Pressures behaviour It can be seen that for a positive clutch flow, there are positive displacements both for the valve plunger and the clutch piston, illustrating the charging phase of the valve when the clutch chamber is filled with oil, while for a negative clutch flow, there are negative displacements, illustrating the discharging phase of the valve and the oil going from the clutch chamber through the valve to the tank.
The behaviour obtained for the valve displacement represented in Fig. 6, is similar with the behaviour obtained in (Balau et al. 2009) where a linearized input-output model and a state-space model were developed for the electrohydraulic pressure reducing valve used as actuator for the wet clutch. The inputs were represented by the magnetic force, 510
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using on/off valves, In: Proc. of 2008 American Control Conference, Seattle, USA, June 11-13. Liao H.-H., M.J. Roelle, and J.C. Gerdes (2008). Repetitive control of an electro-hydraulic engine valve actuation system, In: Proc. of American Control Conference, Seattle, Washington, USA, pp. 957-980. Liu S. and B. Yao (2008). Coordinative control of energy saving programmable valves, In: IEEE Transactions on Control Systems Technology, 16, pp. 34-45. Ma J., G. Zhu, A. Hartsig, and H. Schock (2008). Modelbased predictive control of an electro-pneumatic exhaust valve for internal combustion engines, In: Proc. of American Control Conference, Seattle, Washington, 1113 June, USA, pp. 298-305. Morselli R. and R. Zanasi (2006). Modeling of Automotive Control Systems Using Power Oriented Graphs, In: Proc. of 32nd Annual Conference on Industrial Electronics, IECO? 2006, Paris, France, pp. 5295 – 5300. Peterson K.S., A.G. Stefanopoulou, Y. Wang, and T. Megli (2003). Virtual lash adjuster for an electromechanical valve actuator through iterative learning control, In: Proc. of International Mechanical Engineering Congress, Washington, 15-21 Nov., USA. Tai C. and T.-C. Tsao (2002). Control of an electromechanical camless valve actuator, In: Proc. of American Control Conference, Anchorage, 8-10 May, USA, pp. 262-267. Wang Y., T. Megli, and M. Haghgooie (2002). Modeling and control of electromechanical valve actuator, In: Society of Automotive Engineers, pp. 1-10.
450
Piston displacement [mm]
400 350 300 250 200 150 100 50 0 0.5
1
1.5
2 Time [s]
2.5
3
3.5
Fig. 8. Clutch piston displacement The value obtained for the valve piston displacement is in the range of [-1,+1] mm, like it is supposed to be, because the actuator is a closed loop system and the plunger displacement is restricted by the balance in forces. On the other hand, the clutch is an open loop system, with no feedback, resulting a high value of the piston displacement which can be further limited by designing a proper controller for the electrohydraulic actuated wet clutch. 4. CONCLUSION In this paper a state-space model for an electro-hydraulic actuated clutch used in the automotive control systems for automatic transmission was developed. The model was validated by comparing the results with data obtained on a real test-bench provided by Continental Automotive Romania, which includes a Volkswagen wet clutch actuated by an electro-hydraulic valve. It can be concluded that the simulator has good results illustrated by the similar behaviour obtained for clutch pressure compared with the real measured values.
APPENDIX A. COEFFICIENTS VALUES
ACKNOWLEDGMENT The work was supported by National Center for Programs Management from Romania under the research grant SICONA - 12100/2008. REFERENCES Balau A. E., C. F. Caruntu, D. I. Patrascu, C. Lazar, M. H. Matcovschi and O. Pastravanu (2009). Modeling of a Pressure Reducing Valve Actuator for Automotive Applications, In: Proc. of 18th IEEE International Conference on Control Applications, July 8-10, Saint Petersburg, Russia, pp. 1356-1361. Edelaar M.J.W.H. (1997). Model and control of a wet plate clutch, Ph.D Thesis, Eindhoven. Forrai A., T. Ueda, and T. Yumura (2005). A simple approach to electromagnetic actuator control based on asymptotically exact linearization, In: Archive of Applied Mechanics, 74(8), pp. 550-562. Langjord H., T. A. Johansen and J. P. Hespanha (2008). Switched control of an electropneumatic clutch actuator 511
Symbol
Value/Unit
Symbol
Value/Unit
Ke
1000[N/m]
PT
0[N/m^2]
Mv
25e-3[kg]
w
3e-3[m]
βe
1.6e+9[N/m^2]
VC
7.53e-8[m^3]
ω1 ω2 ω3
1.17e+6[rad/s]
VD
1.04e-7[m^3]
5.42e+7[rad/s]
Vt
3.2e-4[m^3]
8.04e+5[rad/s]
C
3.66e-5[m^2]
ω5
1.04e+6[rad/s]
D
2.94e-5[m^2]
KC =KD
7.58e-11[(m^3/s)/(N/m^2)]
α
2e-5
K1
5.50e-10[(m^3/s)/(N/m^2)]
Mp
0.5
K2
3.52e-9[(m^3/s)/(N/m^2)]
K3
1.26e-8
Kq
5.3418[(m^3/s)/m]
VL
2.51e-5
Kce
2.08e-9[(m^3/s)/(N/m^2)]
Al
7.75e-4 900
Kl
2e-9[(m^3/s)/(N/m^2)]
K
Ps
1e+6[N/m^2]
Bf
0
Ka
0.005
Kb
0.01