Modeling and Control of Continuously Variable Transmissions

Copyright @ IFAC Advances in Automotive Control, Karlsruhe, Germany, 2001

MODELING AND CONTROL OF CONTINUOUSLY VARIABLE TRANSMISSIONS Claus Miiller, Dierk Schroder'

• Chair for Electric Drive Systems , Technische Universitiit Munchen , Germany

Abstract: Transient CVT gear shift behaviour is investigated. Adaptive control laws for partial linearization are proposed. The results allow optimized drive line control. Copyright @2001 IFAC

Keywords: Continuously variable transmission, Adaptive control, Neural networks

management e.g. for reduction or compensation . • In case of conventional or parallel hybrid drive lines, the design of a ratio controller can be optimized. • Torque control of flywheel-based vehicles is potentially improved.

1. INTRODUCTION

Continuously Variable Transmissions (CVTs) will be used in future passenger cars for efficiency and comfort reasons. CVTs allow to choose the engines velocity almost independently of the driving speed. Taking into account the efficiency maps of internal combustion engines, potentials for fuel savings become obvious. While the gear ratio of the CVT is changed, the engine remains coupled to the drive train and is i.g. transmitting torque corresponding to the drivers demand (,Powershift '). Every change of the gear ratio causes an additional torque component in the drive train. This effect is well-known in conjunction with the jerk caused by the gear shifting of automatic transmissions. In order to reduce this jerk in case of an upshift, the engines load is usually reduced by the drive line management corresponding to the ratio transition and the operating conditions.

In this contribution, chain drive CVTs will be investigated. Pushing belt CVTs are characterized by a different construction, but some basic concepts can be transferred as well. The simulation results refer to the drive train of a parallel hybrid passenger car which is discussed as an example for application. It is equipped with a special CVT based gearbox. A set of clutches allows to square the single CVT range of possible gear ratios. A detailed description of the system is given in (Hohn and Pinnekamp, 1994) .

2. MOTIVATION

Hence the CVT can be described as a 'dynamic torque source'. The torque generated by its ratio variation has to be taken into account by the drive train management. For the design of an appropriate controller, the input-output behaviour of the CVT has to be investigated. The benefits of a thorough understanding of the system are:

Controller design for CVT based drive lines is usually (Spijker, 1994) , (Mayer and Schroder, 1998) based on a second order model represented by a two mass system coupled by a time variant gear ratio and a spring-damper element. The inertias of the engine and the vehicle are represented by e 1 and 6 2 respective. Model inputs are the engine 's torque T eng , the rate of ratio variation di/dt and the external torque Text .

• In conventional or parallel hybrid drive lines, the torque components caused by ratio variation can be considered by the power train

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The appropriate control law is easily found using equations (6) and (8). For an effective cancelling of the torque components caused by the CVT , the engine torque has to be influenced in time. Measuring the rate of gear ratio change means considerable effort (Spijker et al., 1992). The gear ratio is usually determined by measuring and dividing the angular velocities cf?1 and rpi. Both signals are measured by inductive pick-ups sensing the pulse pattern generated by slotted discs mounted on the drive shafts or even by the teeth profile of the neighboring gear stages. Depending on the number of slots the signal shows a high noise level. Calculating the rate of ratio change by differentiation requires extensive filtering of the signal, because this operation is very susceptible to signal noise. Filtering causes considerable phase delay.

Fig. l. Drive train model In order to maintain a driveability comparable to a conventional transmission, the wheel torque T2 transmitted by the spring-damper element is chosen as system output. Assuming an elastic two body system without friction the system can be described by the following set of equations:

8 1
R - -(ct: + dE)

= Teng

>000

(1)

TJ

8 2
+ dE) - Text

'00

(2)

.:

Y)lnf

2600

oS

-;2"00

with the torsional angle of the spring-damper element

= 'Pi

f

s: s:

-

2200

(3)

- 'P2

and the gear ratio i and its inverse R

.

rpl rpi

t= -

1 =-

.•

1110~L_~

(4)

R

Time [sI 1100 .~------~----------'

1000

Solving (1) to (4) yields a set of nonlinear differential equations. For a more convenient demonstration of the influence of CVT ratio variation on the wheel torque, in the following section the spring-damper element is assumed to be rigid and no external torques should apply. Then n

_ _~_~_ _,:-_--:-_--:-,:_~.'00

= 1,2

h...-...-NO compensation

~

,I

: \ lOOmS} 10 ms PT,

I I

aoo ;

Filter

I ms

E

bOO-

~

....... 500 ~ ..00 ·

Constant input torque T... Square load function T... : f( <9:)

(5) (6)

o - 25

.. - .-- - - - - - - .-- J

J5

S.5

.t

Timers1

Equation (4) with

rpi = rp2 yields: di


= dt rp2 + i
Fig. 2. Engine velocity / Drive shaft torque system equipped with spring-damper element

(7)

The simulation results shown in figure 2 demonstrate the compensation algorithms effectiveness even in case of a non-rigid system. The torque peak caused by the virtual upshift can be almost compensated, if a PT1-filter with a time constant of 1ms is used. Other filter algorithms with higher stop band attenuation cause higher phase delay leading to instability. Hence measuring is not an option and the gear ratio variation has to be determined by evaluation of the CVT input signal.

With (5) and (6)

T2

=

1 + t 2 TJ 92 91

(-

di

dt

rplTJ811

(8)

Assuming steady state operation at cf?1 and i, T2 is proportional to the rate of gear ratio change di/dt. In order to decouple the engine's angular velocity 01 of the wheel torque T2, a compensating torque component is added to the engine torque.

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The clamping forces are applied hydraulically via the pressures PI ,2' The maximum shift speed is limited by the maximum oil flow Q in the higher pressurized cylinder. The adjustable pulleys axial speed s depends on Q and the piston area A :

3. CVT MODEL Chain drive CVTs consist of three major subsystems. Two pairs of conical sheaves linked by a chain loop form the mechanical transmission. The clamping pressure is determined by two torque sensors. Gear ratio variation is realized by an electro hydraulic valve (EHV) generating a pressure ratio .

. Q

s= -

A

(9)

The kinematics of pulleys and chain deliver a set of three non linear equations with four unknown variables for i = I(s) . Solved numerically, the relation is as shown in figure 4.

3.1 Mechanical Subsystem

Torque respective tension transmission is realized solely by friction between the sheaves conical surfaces and the chains rocker pins. One of the sheaves of each pulley is adjustable so that the chains running radius is determined by axial position and the conical angle of the sheave. In order to minimize losses and prevent chain slip, the axial clamping force has to be chosen corresponding to the transmitted torque. Each steady state operating point characterized by the primary velocity tPl, the gear ratio i and the input torque TI leads to a different equilibrium between the internal forces and the external clamping forces F I ,2. Hence the external forces F1,2 can be distinguished by a torque dependent base level and a operating point dependent relation (= Ft! F 2 .

Axial diJplacemetlt • [nun]

Fig. 4. Kinematics of pulley and chain Due to the complexity of the chain drive, no ex= pression for the input-output-characteristic f(F1,2) can be found. Hence a black-box approach is proposed.

1t

driven

pull
... Fig. 5. Measured (-ratio,

tP = 200 rad/s

Fig. 3. Mechanic transmission (Dittrich, 1953) first introduced a theoretical model of this kind of chain drive. Many publications refer to empirical steady state investigations or descriptions of partial effects. A very extensive dynamic model has been published by (Srnik, 1998) using multibody dynamics. These models are not suitable for control purposes because of their extent and complexity which cannot be discussed in detail here. It can be stated that ( depends nonlinearly on tPI, i and TI as shown in figure 5. During gear ratio shifting, a force relation different to the steady state value of ( is required. In (Ide et al., 1996) the basic dependencies of the shift speed di / dt with respect to the operating conditions are investigated for a pushing-belt CVT. As in the case of steady state operation, no model suitable for control purposes can be given.

3.2 Torque sensors

The clamping forces F1,2 are applied by hydraulic pressures PI and P2' The base level of these pressures is dependent on the transmitted torque in order to prevent chain slip. The pressure is controlled by two torque sensors mounted on the shafts of the CVT . Their working principle is based on a balance between the base pressure and an axial piston force. The axial piston force results of the shaft torque translated by balls rolling on ramps. The base pressure is determined by the oil flow Q and the axial piston position which closes gradually the drain orifice. In (Sauer, 1996) , measurements of the twist angle and the shaft torque can be found. For drive line modeling the

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4. SYSTEM IDENTIFICATION APPROACH

torque sensors have to be modeled by a springdamper element of torque dependent stiffness. ~l

--

_ _ _ ---~

00

.______-

5u

.,/

;;

t.' 0.'

*

An ideal CVT should set the rate of gear ratio variation dependent of the control input UEHV respective the valves position x EHV. The gear ratio would be unchanged = 0 at the middle position, the rate of ratio change would be proportional to the valves deflection. As described above, the chain drive itself and the hydraulics are up to a certain extent non linear. As (Westenthanner , 2000) indicates, a certain pressure difference is necessary to keep the current gear ratio. Any other pressure difference would lead to a ratio variation . Hence the valve position leading to the pressure ratio for the static case must be taken as the new operating condition dependent 'middle position '. As a result, the (-ratio can be treated as an operating point dependent offset Io of the control input I EHV ·

,/

/

Input lorque [Nm]

Fig. 6. Torque sensor characteristic

3.3 Electrohydrnulic valve As depicted in figure 7 both torque sensors are arranged in series. The base pressure is determined by the sensor measuring the higher torque.

*

i,

T ,I{I.

Fig. 9. Nonlinear offset Because the (-ratio is defined for stationary operation only, the valve dynamics can be neglected and the valve position Io can be replaced by the control input offset u o . Hence the original (-ratio

control input

electrohydraulic valve

( = ;~ = f(
constant oil flow (Q)

is transformed by the monotonous valve characteristic in a modified (-ratio

Fig. 7. CVT hydraulic scheme In steady state operation, the line pressures are given by the valve characteristic.

(10)

Pl ,2

(11) The nonlinear behaviour of the chain drive and the valve are summarized by a single non linearity. This way measuring the line pressures or valve position is not required . A ratio controller can be enhanced , by first identifying and then compensating the static nonlinearity ( caused by the chain drive using a neural network . Using the structure shown in figure 10 the offset U o of t he valve position in stationary operation can be identified. The availability of the knowledge gained by the identification structure allows a more accurate control because by adding the identified offset to the control input , the plant is linearized with respect to the (-ratio. On the other hand the difference of the control input and the offset delivers an approximate value for rate of ratio change.

Fig. 8. EHV characteristic The EHV dynamics can be modeled as a first order low pass filter. For proper transmission of small signals, a dither frequency of small amplitude has to be modulated on the input signal. A more elaborate description of the hydraulic subsystem can be found in (Shafai et al. , 1995) .

82

In case of optimal adaption, the estimation error equals zero

i,

with the parameter error (20) which is a constant vector in the input space where

Fig. 10. Identification and compensation structure for stationary operation

./If£ is to be identified.

The nonlinearity is identified using a general regression neural network (GRNN) . This type of neural network allows direct interpretation of the adapted values and shows good extrapolation characteristics.

According to figure 10 di dt = -

A

+ P (ire!

- i)

(21)

di dt

(22)

equation (21) can be written as e=-p . e-~TA

(23)

Since ~ T A is function of time, its frequency domain expression is YP (s) . Then sets) = -p. e(s) - Yp(s)

(12)

e(s) Yp(s)

(13)

= _1_

1+P

= H(s)

(24)

Because the error transfer function H (s) is of PT1 type and hence asymptotically stable (SPR), the system can be compared to error model 3 (Narendra and Annaswamy, 1989) and a Lyapunovbased adapt ion law is given by:

According to (Schroder, 2000) any nonlinearity (14)

(25)

can be expressed by the parameter vector of weights

with TJ denoting the learning step size responsible for the speed of convergence. Using the above adaption law, convergence of parameters can only be guaranteed , if the activation function is persistently exciting. This means that for none of the parameters 1') the activation A may be zero for all time.

and the activation function

In real application of the adaption algorithm in a CVT equipped vehicle this condition will be hardly met, because some areas of the three dimensional input space are not needed for the vehicles operation or even avoided by the drive train management. For this reason the parameter vector e has to be adapted by synthetically generated test pattern at a test rig. The online adapt ion may then be switched off but should be continued in

which is a vector of localized basis functions and a vanishing approximation error d(J;J Using the same concept , the estimate of a nonlinearity ./If£ can be introduced: = N£(q;J = e·T A(~)

• T

+e

e= - -

flil denotes the maximum control deviation allowed for learning. It must be chosen larger than the maximum stationary control deviation caused by the proportional controller gain .

Y

A

With the prerequisite that ire! = const (eqn.12)

In the the following the resulting valve gain is assumed to be 1 and that adaption is permitted only while almost stationary operation:

o

T

T di dt = p. e + ~ A

The control deviation is used as error signal for training the neural network. Utilizing the integrating behaviour of the GRNN a PI-controller is realized. In steady state, the control deviation is zero and the whole control input UEHV is determined by the integral part of the controller. This control input represents the operating point dependent offset.

dire! dt (ire! - i)

e

(17)

The estimation error e(;~J as a result of incomplete adapt ion is given by

83

7. REFERENCES

order to adapt the differences between the actual drive and the reference drive the neural network was trained on . Furthermore the non linearity is considered to be time varying due to wear of the mechanical components and aging effects of the lubricant. These effects can also be adapted by continuing the training process.

Dittrich, O. (1953). Theorie des U mschlingungstriebes mit keilf6rmigen Reibscheibenflanken. PhD thesis. TH Karlsruhe . H6hn , B.-R. and B. Pinnekamp (1994) . Der Autarke Hybrid - ein universelles Antriebskonzept fur PKW. ATZ 96 pp. 294- 300. Ide, T. , H. Uchiyama and R. Kataoka (1996). Experimantal investigation on shift speed characteristics of metal V-belt CVT. In: CVT '96. Yokohama, Japan. pp . 59- 64. ~layer , T . and D. Schr6der (1998). Nonlinear adaptive control of a CVT based parallel hybrid passenger car. In: [FAC Workshop Advances in A utomotive Control. Mohican State Park . Ohio. pp. 115- 121. Narendra. Kumpati S. and Anuradha M. Annaswamy (1989). Stable Adaptive Systems. Prentice Hall International Inc . Englewood Cliffs , New Jersey 07632, USA. Sauer, G. (1996). Grundlagen und Betriebsverhalten eines Zugketten- U mschlingungsgetriebes. PhD thesis. TU Munchen. Schr6der, Dierk (2000). Intelligent Observer and Control Design for Nonlinear Systems. Springer-Verlag. Berlin Heidelberg. Shafai , E. , M. Simons, U. Neff and H.P. Geering (1995). Modelling of a continuously variable transmission. In : IFAC Workshop on Advances in Automotive Control. Monte-Verita.. Ascona, Switzerland. pp. 99-107. Spijker, E. (1994). Steering and Control of a CVT based hybrid transmission for a passenger car. PhD thesis. TU Eindhoven. Netherlands. Spijker, E ., R. van der Graaf, P.J.A Banens and R.F.C Kriens (1992). Measuring the dynamic behaviour of electronically controlled CVT's in hybrid vehicle drive lines . In: Proceedings of the ISATA. Zero em~ssion vehicles electric/hybrid and alternative fuel challenge Conference, ISATA paper 920216 . Florence. Italy. pp. 48-5 2. Srnik . J. (1998). Dynamik von CVT-Keilkettengetrieben. PhD thesis. TU ~Iunchen. 'vVestenthanner. Ulrich (2000 ). Hydrostatische Anpress- und 0bersetzungsregelung fur stufenlose Kettenwandlergetriebe. PhD thesis. TU :-lunchen .

Figure 11 shows the simulated adapt ion result for the modified (-ratio as control input offset.

Fig . 11. Identified

(mod = U o

for


= 200rad/ s.

5 CONCLUSION The input-output behaviour of chain drive CVTs is of interest for drive train control. It is found to be nonlinear due to a variety of effects. For optimal control system design at least knowledge of the static nonlinearities is required . The utilization of neural networks for identification and compensation allows improved ratio control and enables shift speed estimation. 6 NOMENCLATURE

R T) T)

c

d P12

F1.2 (

Q A S

IEHV UEHV

A

e e
primary /secondary velocity [rad / s] primary / secondary torque [Nm] primary/ secondary inertia [kgm 2 ] gear ratio H inverse gear ratio[-] gearbox efficiency [-] learning step size [-] spring coefficient [Nm/rad] damper coefficient [Nm/rad/s] torsion angle [rad] primary / secondary pressure [Bar] primary / secondary clamping force [N] clamping force ratio [-] Oil flow [I/ min] piston surface [m2 ) axial sheave displacement [m] valve position [m] valve control input [-] Activation function [-] Parameter Vector [-] Estimation error [-] Parameter error [-]

84