model based ratio control in a cone ring transmission · 2006-06-29 · model based ratio control...
TRANSCRIPT
Model based ratio control in a cone ring transmission
C.H. Jans
DCT 2006.047
Master Thesis Supervisor: Prof. Dr. Ir. M. Steinbuch Coaches: Ir. T.W.G.L. Klaassen
Dipl. Ing. N. Papakonstantinou (GIF) Technische Universiteit Eindhoven Department of Mechanical Engineering Section Control Systems Technology Eindhoven, June 2006
Erklärung Ich versichere hiermit, dass ich die vorliegende Arbeit selbständig verfasst und keine anderen als die im Literaturverzeichnis angegebenen Quellen benutzt habe. Stellen, die wörtlich oder sinngemäß aus veröffentlichten oder noch nicht veröffentlichten Quellen entnommen sind, sind als solche kenntlich gemacht. Die Zeichnungen oder Abbildungen in dieser Arbeit sind von mir selbst erstellt worden oder mit einem entsprechenden Quellennachweis versehen. Diese Arbeit ist in gleicher oder ähnlicher Form noch bei keiner anderen Prüfungsbehörde eingereicht worden. Aachen, Juni 2006 Casper Jans Geheimhaltung Diese Diplomarbeit darf weder vollständig noch auszugsweise ohne schriftliche Zustimmung des Autors, des betreuenden Referenten bzw. der Firma GIF - Gesellschaft für Industrieforschung mbH vervielfältigt, veröffentlicht oder Dritten zugänglich gemacht werden.
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Table of contents Summary…………………………………………………………….. 3 List of symbols………………………………………………………. 4 Chapter 1: Introduction……………………………………………. 6 1.1 Overview……………………………………………………………………... 7 1.2 Ratio adjustment…………………………………………………………….. 7 Chapter 2: Modelling……………………………………………….. 9 2.1 Servomotor…………………………………………………………………... 9 2.1.1 Dynamic model servomotor………………………………………… 9 2.1.2 Servomotor model validation……………………………………….. 9 2.2 Variator……………………………………………………………………… 11 2.2.1 Geometric relationships…………………………………………….. 11 2.2.2 Ring position adjustment…………………………………………… 11 2.2.3 Variator model validation…………………………………………... 12 2.3 Drill torque disturbance…………………………………………………….. 16 2.3.1 Drill torque calculation……………………………………………... 17 2.3.2 Drill torque model validation……………………………………….. 19 Chapter 3: Control………………………………………………….. 21 3.1 Fuzzy controller……………………………………………………………… 21 3.1.1 Ratio controller……………………………………………………... 21 3.1.2 Servo controller……………………………………………………... 23 3.1.3 High frequency roll-off……………………………………………... 24 3.2 New controller……………………………………………………………….. 24 3.2.1 Ratio controller……………………………………………………... 24 3.2.2 Servo controller……………………………………………………... 25 3.2.3 High frequency roll-off……………………………………………... 28 3.2.4 Ratio controller parameterisation…………………………………… 29 3.2.5 Setpoint limitation…………………………………………………... 29 3.3 Drill torque compensation………………………………………………….. 32 Chapter 4: Simulation and implementation………………………. 34 4.1 Simulation……………………………………………………………………. 34 4.2 Vehicle implementation……………………………………………………... 35 Conclusions and recommendations………………………………... 38 References…………………………………………………………… 40 Appendix A: Figures………………………………………………... 41
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Summary The ratio of the cone ring continuously variable transmission is changed by changing the position of the ring, resulting in different running radii at each of the cones. In the present configuration there is no mechanical restriction of the ring position, only the ring steering angle can be set by a servomotor. Because this setup makes the system marginally stable, a feedback controller is needed. Furthermore, the process is not known fully and is subjected to disturbances. The present controller is a result of many in practice tests of different types of controllers. The parameters of the controller have been determined by intuition and testing. This design method has led to a large number of parameters and dependencies which make controller tuning very time consuming. Furthermore, the performance of the present controller is insufficient in some operating points. Therefore a model based controller with better performance is desired. This report describes the design of such a model based controller. The models made of the relevant components are verified using different methods. Using these models, the present controller is researched which shows a large performance variation. A new optimized ratio controller is designed based on the models which no longer has this variation. By increasing the bandwidth of the servomotor position control loop from 5.5 to 20.8 Hz, the maximum bandwidth of the ratio control loop is increased. The new optimized ratio controller has a constant bandwidth which is higher than the maximum bandwidth of the present controller. Before the new ratio controller is implemented in a vehicle it is tested in a simulation. Both simulation and vehicle tests show the new controller tracks the target ratio much better. Finally some recommendations are made which can further improve the tracking performance if necessary.
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List of symbols cmech Steering mechanism stiffness [Nm/rad] d Process disturbance f Frequency [Hz] hpulse Pulse height [rad] i Variator ratio [-] iplanet Planetary gear set ratio [-] kservo Servomotor constant [Nm/A] ksteer Steering geometry ratio [-] s Laplace variable tdelay Delay time [s] u Process input vtan Tangential speed [m/s] vx Ring speed in x-direction [m/s] x Ring position [m] Cring Ring controller transfer function Cservo Servo controller transfer function Fring Ring controller fuzzy set Fservo Servo controller fuzzy set HCL,servo Servomotor closed loop transfer function Hfilter Noise filter transfer function Hlead Lead filter transfer function HOL,ring Ring open loop transfer function Hring Ring transfer function Hservo Servomotor transfer function Iring Integral ring controller gain Iservo Servomotor current [A]
m2] Jservo Servomotor inertia [kg m2] Jmech Steering mechanism inertia [kg
Lexc Excenter length [m] Llev Lever length [m] Pservo Servo proportional controller gain [Volt/rad] PSring Ring process sensitivity function Rin Input cone radius [m] Rin,0 Minimum input cone radius [m] Rout Output cone radius [m] Rout,0 Maximum output cone radius [m] Rring,in Inner ring radius [m] Rring,out Outer ring radius [m] Rservo Servomotor resistance [Ω] Sring Ring sensitivity transfer function Sservo Servomotor sensitivity transfer function Tdrill Ring drill torque [Nm] Tin Input cone torque [Nm] Tout Output cone torque [Nm] Tservo Servomotor torque [Nm] Uservo Servomotor voltage [Volt]
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α Cone angle [º] ө Servomotor position [rad] φ Ring steering angle [rad] ωin Input cone speed [rad/s] ωout Output cone speed [rad/s] ωring Ring speed [rad/s] ∠ delay Phase delay [º]
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Chapter 1: Introduction Worldwide the demand for continuously variable transmissions (CVT’s) is growing. The most important reason for this is that this type of transmission allows an optimal choice of engine speed. On one hand this can improve fuel economy by keeping the engine speed on the economy line. On the other hand, when maximum performance is desired by the driver, the engine speed can be chosen equal to the speed with maximum power output. For over 10 years, the development aid service company GIF (Gesellschaft für Industrieforschung mbH) has been working on the development of a cone ring transmission (Kegel Ring Getriebe, KRG). The variator of this type of CVT is constructed out of two cones and a metal ring. Advantages of this configuration are high efficiency and high shift rates at low actuator power. The target ratio of the variator results from a drive strategy program, which has two settings, CVT- and AT-mode. In CVT-mode, the variator can have any ratio between the LOW and OD position. In AT-mode, the variator behaves like a conventional stepped automatic transmission, i.e. the variator only operates in a number of fixed ratios. Because in the present configuration the transmission ratio is not fixated mechanically, feedback control is needed to stabilise the system and track the target ratio. The present ratio control strategy is a result of tests in a simulation environment and tests in practise. This controller has multiple parameters which all have to be determined manually by intuition and trial and error. This makes the ratio controller hard to optimise in order to achieve high performance. Also, when a new variator is designed for another vehicle for example, it might be necessary to tune all the parameters again, which is very time consuming. Furthermore, the ratio tracking performance with the current ratio control strategy is not always sufficient. For example, the delay in the first shift of high throttle vehicle accelerations causes too high engine speeds which can be noticed by the driver. Another example is the change in engine speed caused by disturbances at sudden load changes. Although such effects have influence on the vehicle performance and fuel consumption, the effect on the drivability is the most important one. Therefore, an alternative model based controller is desired. The most important feature of this new controller must be that it can be optimised structurally, without the need of many in practise tests. Another requirement is that better tracking performance has to result than with the current ratio controller. However, the tracking performance improvement must not lead to less comfort of the driver. This report describes the design of such a model based controller. In this chapter a general explanation is given, describing the main components of the variator. In Chapter 2 models of these components are presented and validated. These models are used in Chapter 3 to design and optimise an alternative ratio controller. In Chapter 4 the alternative controller is tested in a simulation and implemented in a test vehicle. Finally some recommendations are done to further improve the ratio tracking performance and disturbance rejection.
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1.1 Overview The variator of the KRG is mainly build up by two cones and a ring. Around the first cone, which is connected to the internal combustion engine, a ring rotates and transfers the power to the second cone. To prevent metal to metal contact in the ring inner and outer contact points with the cones, a traction fluid is present. The second cone is connected to the final drive and differential gear. To realise the possibility of reverse driving and vehicle stand still, a planetary gear set and an automated dry clutch are placed between the engine and the input cone. Both cones and the ring are displayed in Figure 1.1. 1.2 Ratio adjustment The ratio of the variator, which depends on the running radii of the cones, is changed by changing the ring position. In order to change the position of the ring, it is steered around an axis orthogonal to both cone surfaces. This steering motion is visualised in Figure 1.1.
Ring
Input cone
Steering mechanism
Output cone
Figure 1.1: Ring steering The ring steering mechanism is depicted in Figure 1.2. An actuator sets the ring steering angle. The ring position is not fixated mechanically. A change in the position of the servomotor results in a small change in the excenter position, because of the planetary gear reduction. The excenter moves the lever which is attached to the control frame. The control frame, sliding guide and ring rotate around the steering axis. The sliding guide is free to move over the control frame.
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Servomotor with encoder
Excenter
Ring Sliding guide
Lever
Control frame
Planetary reduction
Figure 1.2: Ring steering mechanism Only a small steering angle (normally less than 0.5º) and a small actuation torque is needed. The actuator power consumption is small, below 30 W. This setup gives the advantage of changing the ring position quickly with minimal effort. A disadvantage of this setup is the need for constant control of the ring steering angle in order to keep the ring at a constant position.
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Chapter 2: Modelling In order to construct a suitable transmission ratio controller for the KRG, models of the main components are necessary. This chapter describes the design and validation of these models. In addition, a disturbance model, which estimates the drill torque disturbance on the ratio, is presented as well. 2.1 Servomotor First the steering actuator dynamics are described. The position of this DC-servomotor is measured with an incremental encoder. 2.1.1 Dynamic model servomotor Because the electrical dynamics of such a servomotor are much faster than the mechanical dynamics, only the mechanical properties are considered and the electrical are neglected. In addition, the inertia of the adjustment mechanism is neglected because its equivalent inertia is much smaller than the inertia of the servomotor itself. This is because of the high ratio between servomotor position and ring steering angle, the equivalent inertia is given by
2, steermecheqmech kJJ ⋅= (2.1)
The hardware of the servomotor controls the servo armature voltage. The resulting armature current is given by
servo
servoservoservo R
kUI
θ&⋅−= (2.2)
Mechanical damping is neglected because it is expected to have a much smaller effect than the electrical damping. Because the steering mechanism load on the servomotor is hard to model and will probably depend on the operating conditions of the transmission it is not taken into account either. Under these assumptions the following equation of motion holds
θ&&⋅=⋅= servoservoservoservo JIkT (2.3) By combining Eq. (2.2) and Eq. (2.3), this system can be described by
)()()()( 2
servoservoservo
servo
servoservo ksRJs
ksU
ssH+⋅⋅⋅
==θ (2.4)
2.1.2 Servomotor model validation In order to validate the model of the servomotor, a Frequency Response Function (FRF) in closed loop configuration is measured [1]. This method is used in many other control applications as well. The location of the random noise excitation d and plant input u are as indicated in Figure 2.1.
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[x]
Operating point
NoiseExitation
HPlant
CController
d
xu
Figure 2.1: FRF measurement strategy A frequency response function is constructed from the measured time signals d and u using Fourier transformations. The transfer function from d to u is equal to the sensitivity function, given by
)()(11
)()()(
ωωωωω
jHjCjdjujS
servoservoservo ⋅+
== (2.5)
Because the servo controller Cservo is a known function, the measured sensitivity function can be used to determine the plant transfer function Hservo. The measured frequency response is compared in a Bode diagram with the model based frequency response of the servomotor. The model based frequency response is created using the parameters from the data sheet of the servomotor. This Bode diagram is depicted in Figure 2.2. It shows that the model approximates the measured data well.
Figure 2.2: Bode plot FRF measurement and model
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2.2 Variator 2.2.1 Geometric relationships As mentioned before, the variator ratio depends on the position of the ring. Figure 2.3 shows how the running radii of the cones depend on the ring position x.
Rin
Rout
x
α
LOW OD
Rout,0
Rin,0
Figure 2.3: Running radii and ring position The relationship between geometric ratio and ring position of the variator is given by
outring
inring
in
out
outring
inring
in
out
in
out
out
in
RR
xRxR
RR
RR
TT
i,
,
0,
0,
,
,
)sin()sin(⋅
⋅+
⋅−=⋅===
αα
ωω
(2.6)
This relationship is used when converting the measured ring position into a measured ratio, which is used in the ratio control loop. For the steering mechanism displayed in Figure 1.2, the following relationship between the servomotor position ө and the ring steering angle φ holds for small ring steering angles.
steerplanetlev
exc kiLL
⋅=⋅⋅
= θθ
ϕ (2.7)
2.2.2 Ring position adjustment To get insight in the variator ratio dynamics, a model is made of the ring adjustment. Because the position of the ring is only determined by its contact points with the cones, the behavior of such a contact point is investigated. First the outer contact point, between ring and output cone, is described. Assuming zero slip in this contact point, i.e. a pure rolling contact between cones and ring, the speed of the ring results from the steering angle and the tangential speed. Figure 2.4 shows this relationship.
x&
11
φ vtan
vx= x&
φ
Figure 2.4: Outer rolling contact The relation between ring steering angle φ and its speed is given by x&
)tan(tan ϕ⋅== vxvx & (2.8) In which
outoutoutringring RRv ⋅=⋅= ωω ,tan (2.9) Because the ring steering angles remain small (< 2º), Eq. (2.8) can be simplified to
ϕ⋅= tanvx& (2.10) This gives a linear relationship between steering angle and ring speed . The transfer function between the ring position and the steering angle is then given by
x&
ssV
ssXsH ring
)()()()( tan==
ϕ (2.11)
For the inner contact a similar relationship holds. However, a conflict occurs when comparing both relationships. Because of the difference in running radii of the ring in the inner and outer contact point, the outer contact point will always have a higher tangential speed. This means at the same ring steering angle, a larger speed in x-direction will result when evaluating the relationship for the outer contact point. Naturally, the ring can have only one speed in x-direction. Therefore at least one relationship will be incorrect, which means at least one contact will have slip in x-direction. 2.2.3 Variator model validation To investigate if the ring speed in x-direction results from the inner or outer contact point, measurements have been made which compare the measured displacement of the ring with the expected displacement based on each of the two models. During these measurements, the ring steering angle and input cone speed are set at a constant value. To enable large constant ring steering angles for a large amount of time, these measurements were done on a low-speed test rig (maximum input speed 60 rpm). Figure 2.5 shows the results of these measurements.
12
0
20
40
60
80
100
1 2 3 4 5 6 7 8
measurement number
ring
trav
el [m
m]
Inner contact
Measurement
Outer contact
Figure 2.5: Travelled distance in x-direction, measurement and models These results clearly show that for all eight measurements the model based on the inner contact predicts the travelled distance better than the outer contact model. Although the model looks valid for very low speeds, it has to be validated also for more realistic operating conditions. The first approach was to do this by measuring a FRF in a closed loop configuration. A random noise excitation is added to the system input, as depicted in Figure 2.1. Unfortunately this method will not work very well in this application, because of the very small ring amplitudes at higher frequencies. This can be illustrated by calculating the maximum ring amplitude based on the models of servomotor and variator. The amplitude of the servomotor position, which is parallel to the ring steering angle, is limited by the maximum servomotor armature voltage. Figure 2.6 shows the maximum ring amplitude based on the models for a number of realistic values of the tangential speed vtan. The ring position measurement system accuracy of 0.05 mm is displayed as well.
10-1 100 101 10210-4
10-3
10-2
10-1
100
101
102
103
Frequency [Hz]
Am
plitu
de [m
m]
Maximum ring amplitude, different tangential speeds
2 m/s10 m/s40 m/sMeasurement accuracy
Figure 2.6: Maximum ring amplitude Figure 2.6 shows that especially at low tangential speeds and high frequencies the expected amplitude of the ring is too small to be detected by the measurement system, resulting in unreliable results of the FRF measurement. Because a large increase in steering angle amplitude or measurement accuracy is hard to realize, another strategy for validating the ring adjustment model has been used. This strategy inserts a pulse shaped excitation instead of the random noise excitation [2]. The main advantage of this method is the possibility of high amplitudes, because the system is only excited for a short amount of time. This prevents overheating of the servomotor. Furthermore, the relative influence of non-linear effects such as friction and play are also small because of the high amplitude of the excitation signal. Figure 2.7 shows the setup of these measurements and the location of the measured signals, d, u and the unfiltered ring position x.
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i x
[vtan]
Tangential speed
f ie f ie
Steering SystemHCLservo
Pring
Ring controllerCring
PulseExitation
[i_target]
Operatingpoint
f ie x
KRG VariatorHring
x
d
u
Figure 2.7: Pulse measurement strategy ring adjustment The maximum pulse height is limited by the maximum servomotor voltage. When the servo controller output is larger than this maximum voltage, the system will no longer have a linear behavior. When using a proportional controller for the servomotor position, the maximum amplitude of the pulse before the armature voltage saturates can easily be determined. As shown in the diagram in Figure 2.7, the excitation signal is added to the target steering angle. This results in a step in this target value at the beginning of the pulse. Because of the use of a proportional controller, the servo armature voltage will be proportional to the height of the pulse. The following relationship holds for the armature voltage at the beginning of the pulse.
steer
servopulse
steer
servoservo k
Phk
PdU
⋅=
⋅=
ϕ (2.12)
Now the maximum pulse height has been determined, the pulse duration can be determined as well. When the pulse width is too small, the resulting ring position change will also be small and therefore hard to measure. On the other hand, a too large pulse width will lead to large ring movements and the ring will be no longer near the set operating point. In various operating conditions of the transmission (ratio, speed, torque) a large number (>25) of similar pulse responses are measured. The measurement noise on the unfiltered ring position x is filtered offline using forward and backward filtering. This filtering technique has the advantage that it causes no time delay on the signal. After filtering of the ring position signal, the time signals of all pulse responses in one operating condition are averaged. The result of such an average ring position response is depicted in Figure 2.8.
Figure 2.8: Filtered ring position pulse response and average
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In order to compare the measured response with the response of the model, the sensitivity and process sensitivity are used. The sensitivity is the transfer function from d to u and is derived from Figure 2.7. For this setup it is given by
)()()()(
1
1)()()(
tan
,
sVsHsHsCsd
susSringservoCLring
ring ⋅⋅+
== (2.13)
The process sensitivity, which is the transfer function from d to x, is given by
)()()()(
1
)()()()()(
tan
,
,
sVsHsHsC
sHsHsdsxsPS
ringservoCLring
ringservoCLring ⋅⋅
+
⋅== (2.14)
The measured sensitivity Sring and process sensitivity PSring are estimated from the time signals of d, u and x using a state space realisation method [3]. Because the realisation method is based on a normalised step response, the measurement data has to be normalised first. Figure 2.9 shows a comparison between the sensitivity and process sensitivity pulse response from measurement, model and estimation method.
Figure 2.9: PSring and Sring pulse response Figure 2.9 shows a good resemblance between the model based response and the measurements. This also holds for the responses in other operating conditions, which are depicted in appendix A.1. Because the closed loop servo system is also a part of the sensitivity and process sensitivity response, these measurements also show the closed loop model of the servo is valid when the servomotor is in normal operation.
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The step identification method also enables the construction of a Bode plot of the identified variator response. This response can be calculated using the identified sensitivity and process sensitivity state space representations, as well as the modelled servo closed loop transfer function. The identified response of Hring, for frequencies where 0)(, >sH servoCL , can be derived from Eq. (2.13) and Eq. (2.14) and is given by
)()()(
)(, sHsSsPS
sHservoCLring
ringring ⋅
= (2.15)
The Bode plot in Figure 2.10 shows the identified and modelled responses for eight different transmission operating points.
100 10110-2
100
Identified variator Bode plots
mag
nitu
de
modelstep ident
100 101
-140
-120
-100
-80
-60
-40
frequency [Hz]
phas
e [d
eg]
Figure 2.10: Bode plot identified variator Note that no reliable frequency information will be present below a frequency of 0.5 Hz, which corresponds to the data time window of 2 s. Furthermore, because the excitation pulse will have no energy above the frequency corresponding to the pulse width (50 or 20 Hz) [2], the data will be unreliable as well above this frequency. Therefore this Bode plot is constructed from 0.5 to 20 Hz. Although only a small variety of operating conditions have been tested, the measurement data shows the model for the ring adjustment as presented is correct. To be sure the model is correct for the complete operating range of the transmission, in the future a larger number of similar measurements in other operating conditions can be done. 2.3 Drill torque disturbance For the KRG it is known that, because of the non-cylindrical rolling of the ring, in each contact a torque orthogonally to the contact surfaces is generated. These torques are also known as drill torques. The total drill torque on the ring has significant influence on the ratio adjustment, therefore a compensation is designed. Because the rotating directions of cone and ring are equal for the inner contact and opposite for the outer contact, the sign of the inner contact drill torque is opposite to the sign of the outer contact drill torque. The drill torques are represented schematically in Figure 2.11.
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Figure 2.11: Schematic inner and outer drill torques The sum of these two drill torques results in a torque which acts exactly in the ring steering direction. Because the stiffness of the ring and the components of the steering mechanism is finite, this torque will result in a ring steering angle. This unwanted steering angle causes a movement of the ring and therefore a deviation from the target ratio. 2.3.1 Drill torque calculation Here a general discussion of the drill torque model is given. In order to calculate the total drill torque, first the drill torque in each contact point is calculated separately. The calculation method is based on a model which assumes a contact area with small local speed differences between cone and ring. These local speed differences can on one hand be caused by slip in tangential direction, which is needed for torque transfer, and on the other hand by drill speed differences. The drill speed is the speed perpendicular to the cone surface. It can be found by separation of the rotating speed of cone or ring around its rotating axis into a speed parallel and a speed orthogonal to the contact surface. For the output cone this separation is depicted in Figure 2.12.
ωcone
ωdrill ωroll
Figure 2.12: Cone drill speed The drill speed difference is calculated by subtracting the drill speeds of the ring and cone in a contact point. By dividing the contact surface into a large number of elements, a grid can be made of the local speed differences caused by the drill speed difference. The local speed difference resulting from the tangential slip can then be shown in this grid. An example of such a grid is shown in Figure 2.13, without and with tangential slip.
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x-direction
tang
entia
l dire
ctio
n
Velocity profile without slip
x-direction
tang
entia
l dire
ctio
n
Velocity profile with slip
Figure 2.13: Difference speed grids without and with slip The size of the contact surface is determined by a Herzian contact theory which assumes a rectangular shaped contact patch. This theory also calculates the normal force distribution in the contact surface. The local normal force in each element is then converted into an in-plane force using the local friction coefficient. The direction of this in-plane force is assumed to be equal to the direction of the local speed difference. Figure 2.14 once again shows a difference speed grid, the normal force distribution and in-plane force distribution are shown as well. These figures are mere examples, in reality a less course grid is used.
Velocity profile
x-direction
tang
entia
l dire
ctio
n
Normal force distribution
x-direction
tang
entia
l dire
ctio
n
Force profile
x-direction
tang
entia
l dire
ctio
n
Figure 2.14: Local contact speeds and forces All local forces add up to a force and a drill torque. The force is the tangential force which causes power transfer.
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2.3.2 Drill torque model validation To verify the calculations of the drill torques, a number of measurements is desired. Unfortunately it is not possible to measure the drill torques in inner and outer contact separately, because both torques act on the ring simultaneously. Also there is no direct ring drill torque measurement. Therefore, the neutral position of the servomotor is measured. At this neutral position the ring will have no steering angle, i.e. the ring will remain at the same position. To convert the neutral position of the servomotor into a measured drill torque, the drill stiffness of the steering mechanism is used. The relationship between the neutral position and the measured drill torque is given by
mechsteerneutralmechneutraldrill ckdcdT ⋅⋅=⋅= θϕ (2.16) The steering mechanism stiffness has been measured for some components of the adjustment mechanism as well as for the total mechanism. The results of these measurements are shown in Figure 2.15.
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Steering mechanism stifness
drill torque [Nm]
ring
stee
ring
angl
e [d
eg]
ring only
ring + adjustment
complete mechanism
Figure 2.15: Steering mechanism stiffness Figure 2.16 shows the measured drill torque determined from the neutral position of the servomotor and the steering mechanism stiffness. It is compared to the calculated drill torque at different ratios, torques and input speeds.
0 20 40 60 80 100 120-14
-12
-10
-8
-6
-4
-2
0
2Modelled and measured drill torques, i=2.6
input torque [Nm]
drill
torq
ue [N
m]
1500rpm, ModelMeasurement2000rpm, ModelMeasurement4000rpm, ModelMeasurement
0 20 40 60 80 100 120
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2Modelled and measured drill torques, i=1.4
input torque [Nm]
drill
torq
ue [N
m]
Figure 2.16: Drill torque calculation and measurement
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Even though these validation measurements were done on a mechanically different variator, which might not be representative, Figure 2.16 shows that it is difficult to estimate the total drill torque accurately. The main sources of differences between calculated and measured drill torque, in order of their largest effect, are: contact point slip, friction coefficient, contact geometry, steering mechanism stiffness and normal force. Furthermore, the total drill torque is very sensitive to variations in these parameters because the drill torques of the inner and outer contact point are of about equal size but in opposite direction.
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Chapter 3: Control This chapter describes the controllers needed for the transmission ratio. First the old situation is explained and investigated. Then a new, model based, controller design is presented. Finally compensation strategies for the drill torque disturbance are described. 3.1 Fuzzy controller In the old control strategy, a ratio controller sets the target ring steering angle depending on the difference between target and measured ratio. This angle is translated to a target servomotor position using the geometry of the steering mechanism described in Chapter 2.2.1. A second controller makes sure the target position of the servomotor is tracked, setting the servo armature voltage depending on the difference between target and measured position. This strategy results in a cascade structure with two feedback loops. It is displayed in Figure 3.1.
x i
[i_target]
Target ratioU f ie
ServomotorHservo
1
a.s+1Noise filter
Hfil ter
di
i
throttle
I
Integral Ratio controller
f ie x
Hring
di
wring
throttle
Fuzzy
Fuzzy logic Ratio controller
Fuzzy LogicServo controller
Cservo
[throttle]
[wring]
[throttle]
[i ]
Cring
HCL,servo
Variator
Figure 3.1: Fuzzy ratio control structure 3.1.1 Ratio controller As can be seen in Figure 3.1, in the Fuzzy controller, the steering angle of the ring is dependent on the difference between target and measured ratio. This controller consists of a fuzzy logic controller combined with an integral controller. The fuzzy logic controller is a gain scheduled proportional controller. The gain of this controller depends on the ratio tracking error, the measured ring speed and the throttle position. The advantage of this type of controller is that with a relatively small effort and without much knowledge of the system, a reasonable level of performance can be achieved. The gain of the integral part of the controller is scheduled as well. This gain depends on the current transmission ratio and the throttle position. The ratio controller is given by
sthrottleiI
throttleiFsC ringringerrorringring
),(),,()( += ω (3.1)
21
The fuzzy controller parameters have first been determined by simulation tests. Afterwards the controller has been implemented in a test vehicle. Because, in the past, play was present in the steering mechanism, the controller parameters were set weaker to prevent nervous behavior of the system. The dependencies on throttle position and ratio were added to improve the controller behavior during vehicle tests. Because all parameters have been determined by intuition and testing, a large optimization effort is demanded when high tracking performance is required. By using the models described in Chapter 2, the performance of the closed loop system with the Fuzzy control strategy can be investigated. In general, the performance of a controller, evaluated in the frequency domain, can be characterised by its bandwidth, phase and gain margin. The bandwidth of a controller is defined by the frequency where the magnitude of the open loop transfer function crosses 0 dB [4]. This is a measure for the speed of response and disturbance rejection. Phase and gain margin are measures for the robustness of the controlled system. As shown in Figure 3.1, the open loop of the Fuzzy ratio control strategy consists of the ratio controller Cring, the closed loop servo system HCL,servo, the variator and the noise filter Hfilter . The closed loop servo system is given by
)()(1)()(
)(, sHsCsHsC
sHservoservo
servoservoservoCL ⋅+
⋅= (3.2)
The variator consists of two parts, i.e. the model of the ring adjustment Hring (Eq. (2.11)) and the geometric relationship between ring position and variator ratio, given in Eq. (2.6). Because the relationship between ring position and ratio is non-linear, its derivative is needed for the performance investigation. This is given by
20,
0,0,
))sin(()()sin(
αα
⋅+
+⋅−=
xRRR
dxdi
out
outin (3.3)
Using Eq. (3.2) and Eq. (3.3), the open loop transfer function in a operating point is given by
)()()()()( ,, sHdxdisHsHsCsH filter
xringservoCLringringOL ⋅⋅⋅⋅= (3.4)
The phase margin of the ratio control loop is not determined by the dynamics of the components in the open loop alone. In reality, the control algorithm is only evaluated at discrete time intervals, which causes an extra delay. The delay is equal to the time between two intervals, the sample time. In this application the sample time is equal to 0.001 s. This delay causes a frequency dependent additional phase delay given by
delaydelay tf ⋅⋅=∠ 360 (3.5) The influence of the integrating part of the ratio controller is relatively small on the open loop transfer function. This is because its break point is much smaller than the bandwidth. The integrating part is therefore neglected. The applied filter on the measured ratio is taken into account though. It has significant influence, because its breakpoint is near the bandwidth frequency.
22
Due to the different operating conditions of the transmission, a large variation in bandwidth can be detected. This results in a variation of closed loop performance. Because the variation in bandwidth also results in a change in phase and gain margin in this case, the robustness of the controlled system changes as well. The first cause of this large variation can be found in the fact that the ring position reacts faster in case of higher tangential speeds. Some dependency of the tangential speed has been built into the fuzzy controller via the ring speed. However, this dependency is much too small to fully compensate for the behavior of the variator. A second cause for the bandwidth variation can be found in the non-linear relationship between ring position and variator ratio. In LOW an equal change in ring position will result in a much larger absolute change in ratio than in OD (note that OD is the position where i=0.5). This results in a much stronger reaction of the closed loop system in LOW. Figure 3.2 shows the variation in bandwidth and phase margin in different operating conditions of the transmission.
50 km/h
80 km/h 120 km/h
30 km/h
15 km/h
180 km/h
50 km/h
80 km/h 120 km/h
30 km/h
15 km/h
180 km/h
Figure 3.2: Bandwidth and phase margin Fuzzy controller strategy The large variation is clearly visible. At high input speeds in LOW, the controller is on the edge of stability, whereas in OD at low input speeds the setpoint tracking will suffer because of the low bandwidth in those operating points. When trying to increase the bandwidth of the controller in OD by increasing the total controller gain, the phase margin in LOW at high input speeds will become critical and may even result in an unstable controller. 3.1.2 Servo controller Like the controller for the transmission ratio, the controller for the servomotor position is a fuzzy logic controller. It sets the servomotor armature voltage depending on the tracking error of the servomotor position. The output of the controller only depends on this position error. In normal vehicle operation, the position error is such that the controller has constant gain. Hence, the controller acts as a pure proportional controller. The servo controller is given by
)()( errorservoservo FsC θ= (3.6) And for normal operation
servoservo PsC =)( (3.7)
23
Chapter 3.1.1 mentions the importance of the closed loop transfer function of the servomotor control loop for the ratio control. The phase delay of this closed loop is added to the phase delay of the ring adjustment. This means that the bandwidth of the ratio controller is limited by the phase delay of the closed loop transfer function of the servomotor. The Fuzzy servomotor controller has a relative low bandwidth, resulting in a large phase delay and therefore a significant limitation of the ratio controller possibilities. 3.1.3 High frequency roll-off In practise, noise is present in the measurement of the variator ratio. This noise can cause a nervous behavior of the controller and therefore servomotor overheating. To prevent this overheating the measured ratio is filtered. The applied first order filter has a break point at 3.2 Hz (pole at -20 rad/s), causing a significant delay. Because the position of the servomotor is measured with a high resolution discrete encoder, there is no noise on this measurement. Also because there is no differential action in the servo controller, the signal is not filtered. 3.2 New controller In order to solve the problems of the Fuzzy control strategy, a new model based controller is designed. The ratio control structure is changed, but the servomotor position is still controlled separately. The new control structure is depicted in Figure 3.3.
i x[i_target]
Target ratio[vtan]
Tangential speed
U f ie
ServomotorHservo
df ie U
Servo controllerCservo
f ie
Ring controllerCring
1
a.s+1Noise fil ter
Hfilter
f ie x
KRG VariatorHring
Figure 3.3: New ratio control structure 3.2.1 Ratio controller The first problem of the Fuzzy controller is that the gain of the open control loop strongly varies with the tangential speed. Because the tangential speed of the contact points can be calculated continuously from the measured speeds and ring position, the input of the controller can be divided by the measured tangential speed. The linear dependency of this tangential speed of the variator model, together with this division, removes the tangential speed dependency from the open loop. Loosely speaking, the effect of the division by tangential speed is that the ring will get smaller steering angles at higher speeds. As previously mentioned, the non-linear relationship between ratio and ring position also leads to a bandwidth variation of the Fuzzy ratio controller. As can be seen in Figure 3.3, the new ratio controller controls the ring position directly instead of the variator ratio. The ring position control is realised by converting the target and measured variator ratio into a target and measured ring position. This moves the non-linear relationship between ratio and ring position out of the ratio control loop and removes its influence on the closed loop performance.
24
Because of the division by the tangential speed and the control of the ring position instead of the transmission ratio, the open loop transfer function of the ratio control loop has become independent on the operating conditions of the transmission. This results in an equal bandwidth, phase and gain margin in every operating point of the transmission. Controller design is now much easier. For this application a basic PI-controller is used. The integrating part of the controller rejects model inaccuracies and disturbances. The controller has no differential action because no phase lead is required, it also may increase the effect of the expected measurement noise. 3.2.2 Servo controller The gain of the Fuzzy, almost proportional, servomotor position controller has been selected in such a way that the servomotor position will have no overshoot on a step in its target value. This is equivalent to a damping of 100% and a phase margin of approximately 75º. In order to create more possibilities for the ratio controller, the bandwidth of this controller has to be increased. The choice has been made to maintain this property for the new servo controller as well. To increase the bandwidth of this controller without decreasing phase margin, phase increase is needed. A well known method for adding extra phase is a lead filter. The transfer function of such a lead filter is given by
12
1
12
1
+⋅⋅⋅
+⋅⋅⋅
=s
f
sf
H
pole
zerolead
π
π (3.8)
This filter has a pole at high frequency in order to reduce the effect of attenuating noise at high frequencies. The parameters of the new servo controller, the pole and zero frequency of the lead filter and the controller gain, are determined using a root-locus method [4]. This method has been selected because it can easily give insight when the damping of a system is 100%. The speed of response of the controller can also been read out by looking at the shortest pole distance to the origin. This shortest pole distance is similar to the bandwidth of a controller. The goal is to place the poles of the controlled system as far from the origin as possible, while keeping all poles on the negative real axis. This is comparable to a stable system with maximum bandwidth at 100% damping. Figure 3.4 shows a root-locus plot of the old servo control loop. The poles of the servomotor itself result from its transfer function (Eq. (2.4)) and are equal to 0 and -139 rad/s
( 1392
=⋅ servoservo
servo
RJk
).
25
Figure 3.4: Root locus plot old servo control loop From Figure 3.4 can be concluded that increasing the controller gain will eventually lead to pole locations off the negative real axis and thus to a damping less than 100%. When the gain of the Fuzzy servo controller is evaluated, it shows that it is indeed maximised at 100% damping. Both poles are at the break-away point which results in a double smallest pole
length of 70 rad/s ( 702
2
=⋅⋅ servoservo
servo
RJk
).
In order to maximise the smallest pole length, the zero of the lead filter is placed almost on the negative real pole of the servomotor. The lead filter pole is placed on the negative axis by a factor 4 from the zero. Figure 3.5 displays a root locus plot of the servo control loop with the lead filter.
Figure 3.5: Root locus plot new servo control loop To select the correct controller gain, the damping and pole distances from the origin are plotted as a function of this gain in Figure 3.6. It clearly shows at which gain the damping of the control loop becomes less than 100%. In this case the maximum gain is 2.92 Volt/rad. For illustration purposes, the pole damping and pole lengths of the Fuzzy controller are displayed as well.
26
Figure 3.6: Pole damping and pole lengths It can be seen in Figure 3.6 that with this lead filter, a much higher gain can be selected at 100% damping, i.e. the bandwidth can be higher. Now the parameters of the new controller are known, the Bode plots of both servo controllers can be compared. Figure 3.7 shows the Fuzzy controller having a smaller gain and no phase lead whereas the new controller has a phase lead of 36.9º at 44 Hz. When the open loop Bode plot is looked at, a significant higher bandwidth (20.8 vs. 5.5 Hz) can be noticed while maintaining equal phase margin.
10-1 100 101 102 103
100
101
mag
nitu
de
Servo controller Bode plots
10-1 100 101 102 103-10
0
10
20
30
40
frequency [Hz]
phas
e [d
eg]
OldNew
10-1 100 101 102 10310
-5
100
105
mag
nitu
de
Open loop Bode plots servo
10-1 100 101 102 103
-180
-135
-90
-45
frequency [Hz]
phas
e [d
eg]
OldNew
Figure 3.7: Bode plots controllers and open loop servo The increased bandwidth results in a significant improvement of the closed loop phase delay. Both closed loop Bode plots are displayed in Figure 3.8.
27
10-1 100 101 102 10310-4
10-2
100
mag
nitu
deClosed loop Bode plots servo
10-1 100 101 102 103
-180
-135
-90
-45
0
frequency [Hz]
phas
e [d
eg]
OldNew
Figure 3.8: Closed loop Bode plots servo control loop As mentioned before, the servomotor closed loop phase delay is important for the properties of the ratio control loop. The decreased phase delay enables a significant increase of the ratio controller bandwidth. If in the future the closed loop phase delay is still considered too large, the bandwidth of the controller can easily be increased by allowing less than 100% damping. 3.2.3 High frequency roll-off As mentioned in Chapter 3.1.3, the currently used ratio measurement filter causes a large phase delay. This phase delay decreases the phase margin of the ratio control loop and with that the robustness and maximum bandwidth of the ratio controller. Therefore, the filter breakpoint is moved to 15.9 Hz (pole at -100 rad/s) instead of 3.2 Hz. This significantly improves the phase characteristic of the filter as shown in Figure 3.9. Furthermore, the filter is placed on the ring position instead of the variator ratio, to give a linear behavior in the control loop.
10-1 100 101 102 103
10-2
100
mag
nitu
de
Noise filter Bode plots
10-1 100 101 102 103
-90
-45
0
frequency [Hz]
phas
e [d
eg]
OldNew
Figure 3.9: Bode plots measurement filters The weaker filter may result in more nervous behavior of the controller. However, practical application shows that this does not lead to servomotor overheating.
28
3.2.4 Ratio controller parameterisation Now the transfer functions of the filter and the closed servo loop are known, the parameters for the ratio controller can be determined. The bandwidth of the ratio controller is increased until its phase margin reaches 45º. This value is often used in control applications. The integral controller gain is determined by a rule of thumb. It states that the zero location of the PI-controller should be equal to 20% of the bandwidth. The result of the parameter determination is shown in Figure 3.10. First the Bode plot of the controller is given, then the Bode plot of the open loop transfer function of the ratio controller is shown.
10-1 100 101 102 103101
102
103
mag
nitu
de
Bode plots new ratio controller
10-1 100 101 102 103
-90
-45
0
frequency [Hz]
phas
e [d
eg]
10-1 100 101 102 10310-10
10-5
100
105
mag
nitu
de
Open loop Bode plots ring position
10-1 100 101 102 103
-180
-135
-90
frequency [Hz]
phas
e [d
eg]
Figure 3.10 Bode plots new ratio controller and open loop When comparing Figure 3.10 and Figure 3.2, clearly the bandwidth with the new controller (5.3 Hz) is larger than the highest bandwidth with the Fuzzy controller. Furthermore, the performance of the closed loop system no longer depends on the operating conditions of the transmission. 3.2.5 Setpoint limitation Because the damping of closed loop system is less than 100%, overshoot will occur when the ratio target value is faster than the physically possible ratio adjustment. This overshoot is an undesired effect. To further illustrate this problem, the response of the controller on a setpoint step is investigated. Physical limits on the system make an exact tracking of this setpoint impossible. Because the setpoint step suddenly causes a large error between target and measured ratio, the controller output will be very large. In spite of the maximum effort of the controller to track the setpoint step as quickly as possible, the tracking error remains large in the first period. This error over this amount of time causes the contents of the integrating part of the controller to windup. When the target ratio is eventually met, the contents of the integrating part of the controller still results in a positive steering angle. The nature of the integrating part is such that its contents can only decrease in case of a negative error. The negative error needed for the integrating part to decrease its contents is also known as overshoot. This is illustrated by Figure 3.11, in which the windup of the integral part is shown schematically.
29
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
1.2
1.4
time [s]
setpointsystem response
Integral windup
Integral windoff
Figure 3.11: Integral windup at step response To prevent unnecessary windup of the integral part of the controller, the best method is to use a ratio setpoint which can always be tracked physically. When the controller output increases in such a case, the control error will disappear and the windup of the integral part will be stopped. Depending on the order of the system, for example a second or third order setpoint describing the fastest possible movement to the target position is generated. This technique is applied in many point-to-point motion control systems [5]. An example of such a second order setpoint signal is displayed in Figure 3.12. In this example the maximum acceleration and speed of the system are limited.
0 0.5 1 1.5 2 2.5 3
0
0.2
0.4
0.6
0.8
1
time [s]
original2nd order
Maximum acceleration
Maximum speed
Maximum deceleration
Figure 3.12: Second order setpoint signal Such a setpoint is generated using knowledge of the maximum performance of the components of the system, such as maximum acceleration and speed. Most point-to-point systems have requirements stating when they should be at which position. This immediately reveals the problem for application of such a setpoint generation for the KRG. In reality, the setpoint future is unknown. The setpoint generation uses operating conditions such as vehicle speed, engine speed and accelerator pedal position. These are determined by road conditions and the driver, they are therefore unknown in advance. Therefore, an alternative method has to be found which limits the raw setpoint signal in such a way that a setpoint results which can always be tracked physically. For the KRG there are three important limits for the ratio adjustment. The first limit is the maximum mechanical steering angle of the ring. Because this steering angle is parallel to the ring speed in x-direction, a first order limit results on the ring position setpoint. The next equation shows the relationship between maximum speed in x-direction and maximum steering angle.
maxtanmax ϕ⋅= vx& (3.9)
30
The second limit is the maximum speed of the servomotor. This speed is limited by the maximum servomotor armature voltage and is parallel to the ring steering angle speed, which is parallel to the ring position acceleration. The relation between the maximum ring acceleration and servomotor speed is given by
maxtanmaxtanmax θϕ &&&& ⋅⋅=⋅= steerkvvx (3.10) The third limit is the maximum acceleration of the servomotor. This acceleration depends on the armature voltage as well as the momentary speed of the servomotor, as can be concluded from Eq. (2.2) and Eq. (2.3). The servomotor acceleration is parallel to the third order derivative of the ring position according to
maxtanmax,tanmaxtanmax θωϕ &&&&&&&& ⋅⋅=⋅⋅=⋅= steerservosteer kvkvvx (3.11) The equations above show that each limiting factor depends on the tangential speed. This variable has to be taken into account in the design of a setpoint limiting algorithm. A first order setpoint limiter on the ring position is fairly easy to realise. The strategy is to move towards the target ratio as fast as possible, until the setpoint is met and the setpoint speed is smaller than the maximum speed. The maximum speed can be calculated continuously from the maximum steering angle and the measured tangential speed. The strategy is illustrated in Figure 3.13.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
time [s]
originalrate limited
Limiter active
Limiter inactive
Figure 3.13: First order setpoint limiting The limiting of the second order derivative of the setpoint is somewhat harder. Because the speed of the setpoint cannot change instantaneously, the maximum acceleration towards the setpoint has to be stopped sooner. In order to reach the setpoint exactly and as soon as possible, the calculated stopping distance is used. The stopping distance is the minimum distance needed to come to a stop from a certain speed and can be calculated using
max
2
max
2
22 xx
avxstop &&
&
⋅=
⋅=Δ (3.12)
This stopping distance is calculated continuously from the speed of the limited setpoint. As soon as the distance between the raw and limited setpoint becomes smaller than this stopping distance, the strategy switches to maximum deceleration. The resulting limited setpoint signal is displayed in the Figure 3.14.
31
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.2
0.4
0.6
0.8
1
time [s]
original2nd order limited
Limiter active
Limiter inactive
Figure 3.14: Second order setpoint limiting A robust algorithm for third order limiting has not been found yet. Because of the high acceleration limit of the servomotor and its dependency on its momentary speed, the use of a second order limiter is accepted. In the future the use of a third order limiter is recommended to make sure the setpoint can exactly be tracked physically. 3.3 Drill torque compensation The steering mechanism stiffness has been described in Chapter 2.3.2 and is equal to 25 Nm/º. When using the maximum value of the drill torque calculation of approximately 10 Nm a maximum steering angle disturbance of 0.4º results. This value may seem small, but is very large when compared to a typical ratio adjustment steering angle of 0.1º. The effort in creating a drill torque compensation will therefore probably pay off. The main advantage of this disturbance compensation is an improvement of the ratio controller performance without changing its bandwidth and robustness. To compensate for the effect of the total drill torque, two strategies are presented. The first strategy is to calculate the resulting steering angle by using models of the drill torques and the steering mechanism stiffness. The calculated steering angle can then be used to give an opposite signal to the target position of the servomotor. The servomotor will then steer the ring in the opposite direction, cancelling out the steering angle caused by the drill torques. Figure 3.15 shows how this compensation strategy can be added to the control loop.
i x[i_target]
Target ratio[vtan]
Tangential speed
U f ie
Servomotor
df ie U
Servo controller
C
Ring controller
Ratio
Input torque
Slip
Operatingconditions
1
a.s+1Noise fi lter
Hfilter
f ie x
KRG Variator
Tdrill f ie drill
Elasticitymodel
Tdrill
Dril l torquemodel
f ie
Drill steering angle
Figure 3.15: Drill torque compensation strategy The second compensation strategy uses the measured change in neutral position of the servomotor directly, and therefore does not require an accurate calculation of the drill torque. It also diminishes the need for an accurate steering mechanism stiffness determination. This compensation strategy only needs reliable measurements of the neutral position of the servomotor, in a large number of operating points of the transmission.
32
Although the second method will probably give the most accurate prediction of the required steering angle correction, the large number of reproducible measurements is a big disadvantage. Firstly, the number of measurements will be very large because ratio, speed and input torque can be varied. Secondly, in case of a mechanical change in for example the ring geometry, the complete measurement cycle has to be done again. Another disadvantage of this strategy is that no extra insight in the generation of the drill torques results. Because the development of the drill torque calculation is still in progress and a large number of neutral position measurements is undesired, a drill torque compensation will not be implemented for this assignment.
33
Chapter 4: Simulation and implementation Now a new ratio controller has been designed, its performance can be tested and compared to the Fuzzy controller. Before the new ratio controller can be implemented in a test vehicle it is tested in a simulation environment. This enables a fast and safe check of how the new controller will behave. After satisfactory simulation results, the new controller is implemented in a vehicle. Several driving tests have been done to compare both ratio controllers. 4.1 Simulation The simulation model is a complete drive train model, including the KRG as described in Chapter 1, power transfer in the contact points, a model of the combustion engine and vehicle mass. Furthermore, a full generation of the target ratio is present which is exactly equal to that of the vehicle. As far as the ring adjustment system is considered, realistic limits such as maximum mechanical steering angle and servo armature voltage are built in. To compare both controllers, a driving cycle has been designed in which some extreme driving conditions are built in, such as a kickdown and an emergency brake. The driving cycle sequence is as follows:
1. Medium brake from 100 km/h until vehicle speed is under 20 km/h 2. Accelerator pedal 10% for 5 seconds 3. Accelerator pedal 100% until vehicle speed is over 120 km/h 4. Medium brake until vehicle speed is under 60 km/h 5. Accelerator pedal 20% for 5 seconds 6. Emergency brake until vehicle speed is under 10 km/h 7. Accelerator pedal 50% until vehicle speed is over 100 km/h
This cycle is carried out for the CVT-mode as well as the AT-mode of the transmission. The results are compared in two different ways. First the time signals of the target and measured value of the transmission ratio are compared. This can be seen in Figure 4.1.
0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
3
time [s]
ratio
[-]
CVT simulation results
Target ratioOld controllerNew controller
Kickdown Brake
LOW
OD
0 10 20 30 40 50 600
0.5
1
1.5
2
2.5
3
time [s]
ratio
[-]
AT simulation results
Target ratioOld controllerNew controller
Kickdown Brake
LOW
OD
Figure 4.1: Simulation results drive cycle CVT and AT
34
To give more insight in the performance of both controllers, so called variograms are constructed. These show the vehicle speed as a function of engine speed. The difference between target and measured engine speed can easily been read out from these figures. These errors are important in reality because of the requirements on vehicle acceleration, fuel consumption and comfort for example. The variogram of the kickdown situation (Figure 4.2) is constructed using the same data as in Figure 4.1.
0 1000 2000 3000 4000 5000 60000
50
100
150Variogram CVT kickdown simulation
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetOld controllerNew controller
LOW
OD
0 1000 2000 3000 4000 5000 60000
50
100
150Variogram AT kickdown simulation
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetOld controllerNew controller
1st gear
2nd
3rd
4th
5th 6th 7th
Figure 4.2: Variogram kickdown simulation CVT and AT The simulation results show a large difference between performance of the Fuzzy and the new ratio controller. The new ratio controller has superior ratio setpoint tracking in all tested conditions. 4.2 Vehicle implementation To investigate if the new controller works as well in practice as it does in simulation, it is implemented in a test vehicle. Some care has to be taken before the division by the tangential speed can be implemented though. First, the minimal measured tangential speed has been limited to 5 m/s, corresponding to approximately 15 km/h vehicle speed. This is done to prevent nervous behavior and large steering angles at low vehicle speeds. Because a speed sensor fail can lead to a measured tangential speed lower than the actual tangential speed an unstable situation may result. To prevent this from happening, the tangential speed signal, which is normally calculated from the output cone speed and the output cone running radius, is backed up by a signal which uses the measured wheel speed and the output cone running radius. A backup signal which uses the input cone speed and the input cone radius can also be used. Before the controller parameters are set to their full strength, a number of tests are made with weaker parameters. After these tests, which are done for safety, the controller parameters are set to their normal values. Then screenings are carried out. Such a screening is a vehicle launch with a fixed pedal position. These screenings are done at different accelerator pedal positions in CVT-mode as well as in AT-mode.
35
Similar to the results of the simulation, the target ratio and measured ratio are plotted against time. Variograms are constructed as well, which show three different signals. The first signal is the target engine speed, which is calculated from the measured output cone speed and the target variator ratio. The second signal is called the geometric engine speed, it is calculated using the measured output cone speed and the measured geometric ratio. The real measured engine speed is displayed as well. The difference between real engine speed and geometric engine speed is caused by slip in the contact points. Because the geometric ratio is controlled, the difference between target and geometric engine speed is caused by the ratio tracking error. Figures 4.3 and 4.4 show the results for a 50% pedal position in AT-mode and a 80% CVT screening. Similar figures for 30% CVT, 50% CVT, 30% AT and 80% AT can be found in appendix A.2.
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 50% AT mode, old controller
TargetMeasured
0 500 1000 1500 2000 2500 3000 3500 40000
10
20
30
40
50
60
70
80
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
] TargetGeometricReal engine speed
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 50% AT mode, new controller
0 500 1000 1500 2000 2500 3000 3500 40000
10
20
30
40
50
60
70
80
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure 4.3: Vehicle screening 50% AT
36
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 80% CVT mode, old controller
TargetMeasured
0 1000 2000 3000 4000 50000
10
20
30
40
50
60
70
80
90
100
110
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetGeometricReal engine speed
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 80% CVT mode, new controller
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000
10
20
30
40
50
60
70
80
90
100
110
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure 4.4: Vehicle screening 80% CVT The screening results show a significant increase in performance. The large delay during the first shift causing too high engine speeds has been removed. Unlike with the Fuzzy controller, the ratio tracking error now results in a smaller deviations in engine speed than slip does. Loosely speaking, this means a further improvement in ratio tracking performance will have only small influence on tracking the target engine speed. When observing the new controller data closer, some small oscillations on the measured ratio can be seen in the LOW range. Even though these oscillations are not noticeable in the vehicle, they are undesirable. The cause of the oscillations may lie in a deviation between the real variator and its adjustment model. This deviation can cause a smaller phase margin which means less damping. A controller parameter optimization might solve this problem.
37
Conclusions and recommendations Conclusions By designing the controller based on models of the relevant variator components, the parameters can be determined structurally, without the need of many time consuming tests. To justify the use of these models, they have been verified by use of a FRF and a pulse identification method. By dividing tangential speed out of the control loop and controlling the ring position instead of controlling the variator ratio directly, the behavior of the controlled ratio system has been made constant over its whole operating range. This enables the use of a simple PI-controller with constant parameters. To increase the robustness and the possible bandwidth of the ratio controller, the bandwidth of the servomotor position controller has been increased. The parameters of the ratio controller are determined using basic controller design rules. After good results of the new controller in simulations it has been implemented in a test vehicle. The screening measurement results show that the new controller performs significantly better. Also a method for compensating drill torque disturbances has been presented. This can have a large effect, because of the small steering angles needed for ratio adjustment and the relatively low stiffness of the adjustment mechanism. Recommendations A number of recommendations are stated below which can result in a better performance of the controller in the future. Parameter optimization The present new controller parameters have been determined using rules of thumb. Because small oscillations can be seen in the measured ratio signal of the vehicle screening tests, an optimization of the parameters of the new controller is desirable. This can be done by creating better adjustment mechanism models, by doing more identification measurements. Setpoint limiting As mentioned in Chapter 3.2.5, setpoints which cannot be tracked physically can cause problems when implementing high bandwidth controllers. The present second order setpoint limiter only limits the acceleration of the ring position setpoint. This limit is proportional with the maximum rotation speed of the servomotor. Because this speed cannot be reached instantaneously, a limiter should be implemented which also accounts for the maximum acceleration of the servomotor. The construction of such a limiter will be complex because the maximum servomotor acceleration depends on its momentary speed. Furthermore, a robust third order limiter for an unknown signal is hard to design.
38
Setpoint feed forward When the performance of the ratio controller is not satisfying, it can be increased by using a feed forward strategy on the ratio setpoint signal. The feed forward strategy means the steering angle of the ring will be set to the value where it is equal to the required steering angle. This angle is based on the model of the ratio adjustment and the speed in x-direction of the setpoint signal. The speed of the setpoint signal can be determined by differentiation of the target ring position signal, but this may give numerical problems. Another possibility is to use a signal from the setpoint limiter described in Chapter 3.2.5, this signal will be more robust than a numeric differentiation of the setpoint signal. The main advantage of this feed forward strategy is that the ring is steered at the correct steering angle as soon as a change in target ratio occurs, whereas without feed forward, a difference between target and measured ratio has to be present to steer the ring. Because this difference is particularly large for fast ratio adjustments, feed forward may be considered in the future. Drill torque compensation The total drill torque causes a significant disturbance of the ring steering angle. Therefore a compensation will probably significantly improve the performance of the controller. However, the present calculation of the drill torque does not seem accurate enough to fully compensate this effect. For better results the calculation accuracy has to be increased. An accurate model of the steering mechanism stiffness is important as well. The validation of the drill torque calculation requires reliable measurements. The data used in this report has only been measured once via the neutral position of the servomotor. In the future a more direct reproducible measurement of the drill torque is desired. If the drill torque calculation cannot be improved sufficiently, reproducible measurements of the neutral position of the servomotor can be used. Unfortunately, these measurements are quite time consuming. Ratio controller bandwidth When a higher ratio controller bandwidth is desired, this can be realised by increasing the servomotor position controller bandwidth. A better ratio measurement can also be looked at, decreasing the need for strong filtering. Component mechanics In spite of assumptions of linear relationships for the controller design, some non-linear effects such as friction and play will be present in the steering mechanism as well as in the ratio measurement system. The influences of such effects should be investigated.
39
References [1] J. Boot, Frequency response measurement in closed loop: brushing up our knowledge.
TU/e, DCT 2003.59, Eindhoven, April 2003 [2] T.W.G.L. Klaassen, M.Steinbuch, Identification and Control of the Empact CVT.
Submitted to IEEE Transactions on Control Systems Technology, May 2006. [3] B. De Schutter, Minimal state-space realization in linear system theory: An overview.
Journal of Computational and Applied Mathematics, Special Issue on Numerical Analysis in the 20th Century – Vol. I: Approximation Theory, vol. 121, no 1-2, p. 331-354, Sept. 2000.
[4] Gene F. Franklin, J. David Powell, Abbas Emami-Naeini, Feedback control of
dynamic systems. Addison-Wesley, New York, June 1995. [5] P. Lambrechts, Trajectory planning, setpoint generation and feedforward for motion
systems. Lecture sheets Digital Motion Control 4K410, TU/e, Eindhoven 2003 [6] S.W.H. Simons, Reducing clamping forces in the cone ring transmission. TU/e, DCT
2004.118, Eindhoven 2004
40
Appendix A: Figures This appendix shows results of tests which are done in other operating points than the figures displayed in the text. A.1 Validation ring adjustment model Figures A.1 to A.9 show the measured, identified and modelled pulse response for sensitivity and process sensitivity of the ring adjustment, as mentioned in Chapter 2.1.3. First an overview is given of the settings for each measurement. Measurement Ratio [-] Speed [km/h ] Torque [Nm] Pulse width [ms] Pulse height [rad]
A 1.3 30 0 50 0.008B 1.3 30 100 50 0.008C 1.3 60 0 20 0.008D 1.3 60 75 20 0.008E 1.8 22 0 50 0.008F 1.8 22 100 50 0.008G 1.8 44 0 20 0.008H 1.8 44 75 20 0.008
Table A.1: Measurement legend pulse response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement A
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement B
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
Figure A.1: Results measurement A Figure A.2: Results measurement B
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement C
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement D
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
Figure A.3: Results measurement C Figure A.4: Results measurement D
41
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement E
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement F
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
2
3x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
Figure A.5: Results measurement E Figure A.6: Results measurement F
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement G
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10
15
20x 10-4 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, measurement H
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10
15
20x 10-4 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
Figure A.7: Results measurement G Figure A.8: Results measurement H
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5
0
5
10x 10-3 Sensitivity (u) pulse response, large deviation
refe
renc
e rin
g an
gle
[rad] step ident
measurementmodel
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-4
-2
0
2
4
6x 10-3 Process sensitivity (x) pulse response
time [s]
ring
posi
tion
[m]
Figure A.9: Large model deviation Figure A.9 displays a measurement with a large deviation between model and measurement data. This can to some extent be explained by the uncommon large amount of slip in the variator in this operating point. The large amount of slip may result in a non-rolling contact between cones and ring which results in a smaller adjustment in x-direction. This effect can be compared with a car which becomes uncontrollable when the steered wheels are slipping.
42
A.2 Controller implementation in vehicle Here the performance of the Fuzzy and new controller is compared when implemented in a test vehicle. Because the results of the 50% AT-mode and 80% CVT-mode are already shown in Chapter 4.2, only the results for 30% CVT, 50% CVT, 30% AT and 80% AT-mode will be displayed.
0 2 4 6 8 10 120.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 30% CVT mode, old controller
TargetMeasured
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
30
35
40
45
50
55
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetGeometricReal engine speed
0 2 4 6 8 10 120.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 30% CVT mode, new controller
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
30
35
40
45
50
55
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure A.10: Vehicle screening results 30% CVT
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 50% CVT mode, old controller
TargetMeasured
0 500 1000 1500 2000 2500 3000 3500 40000
10
20
30
40
50
60
70
80
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetGeometricReal engine speed
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 50% CVT mode, new controller
0 500 1000 1500 2000 2500 3000 3500 40000
10
20
30
40
50
60
70
80
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure A.11: Vehicle screening results 50% CVT
43
0 2 4 6 8 10 120.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 30% AT mode, old controller
TargetMeasured
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
30
35
40
45
50
55
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetGeometricReal engine speed
0 2 4 6 8 10 120.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 30% AT mode, new controller
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
30
35
40
45
50
55
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure A.12: Vehicle screening results 30% AT
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 80% AT mode, old controller
TargetMeasured
0 1000 2000 3000 4000 50000
10
20
30
40
50
60
70
80
90
100
110
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
TargetGeometricReal engine speed
0 1 2 3 4 5 6 70.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
time [s]
ratio
[-]
Screening 80% AT mode, new controller
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 55000
10
20
30
40
50
60
70
80
90
100
110
Engine speed [rpm]
Veh
icle
spe
ed [k
m/h
]
Figure A.13: Vehicle screening results 80% AT
44