Jurnal Mekanikal 2008 BS KBT-2

8/6/2019 Jurnal Mekanikal 2008 BS KBT-2

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PD CONTROL PERFORMANCE FOR RATIO CONTROL OF AN

ELECTROMECHANICAL DUAL ACTING PULLEY (EMDAP- CVT)

SYSTEM

B. Supriyo, K.B. Tawi, H. Jamaluddin & S. AriyonoFaculty of Mechanical Engineering, Universiti Technologi Malaysia, 81310 UTM Skudai, Johore, Malaysia

ABSTRACT

This paper proposes proportional derivative (PD) controller for transmission ratio control of Electro-

mechanical Dual Acting Pulley (EMDAP) CVT system. The initial values of the PD gains are derivedusing Astrom and Hagglund tuning method involving a closed-loop relay feedback experiment to attain

both critical waveform oscillation period and critical gain. Based on these critical values, both

proportional and derivative gains can be obtained using Ziegler-Nichols formula. Then, the proportional

gain is experimentally fine tuned to improve the system performance in terms of less overshoot, smallersteady state error and smaller trajectory error. Experimental results for no load torque condition are

presented to demonstrate the effectiveness of the proposed controller.

Keywords: Electromechanical CVT, Dual Acting Pulley CVT, Ratio Control, PD Controller.

1.0 INTRODUCTION

Continuously Variable Transmissions (CVTs) have become very popular in automotive applications.

Compared with a manual transmission or a stepped automatic transmission, CVT has wider range of

transmission ratio coverage. This unique characteristic of CVT makes it possible for engine operating

conditions to be adjusted accordingly to achieve its minimum fuel consumption or maximum engineperformance. Most of current CVTs employed hydraulic actuator. This kind of CVT requires continuous

power from the engine to supply the required hydraulic force to maintain its desired transmission ratio and

to prevent belt slip. It is well known that this continuous power consumption causes the major loss in the

hydraulic CVT system, hence reducing CVT efficiency [1]. In addition, these CVTs are designed with

single acting pulley mechanism. It means, that only one pulley sheave in each pair is movable, henceintroducing belt misalignment. Long term application of this misalignment may damage the belt. This belt

misalignment has been studied intensively in [2].

This paper introduces an electro-mechanical dual acting pulley CVT (EMDAP CVT) system with two DC

motors as its actuators. This actuator works only during transmission ratio changes, hence shortening

actuators operation time and reducing energy loss. As compared to the Electro- mechanical CVT

proposed in [3] that employs one movable pulley sheave, the EMDAP CVT adopts two movable pulleysheaves in each shaft to eliminate belt misalignment. Each pair of movable sheaves in each shaft is driven

by DC motor system. The primary motor is used for changing the CVT ratio, while the secondary one is

used for preventing the belt from slipping.

PID (Proportional, Integral and Derivative) controller has been the basis in simple linear control systems

for many years. The PID controller is a well-known technique for various industrial control applications,

mainly due to its simple design, straightforward parameters tuning and robust performance. PID

controllers are very common for motor control applications [4, 5, 6 & 7]. For motor position control


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applications, PD controller is the most popular. Therefore, this paper proposes a PD controller forcontrolling the EMDAP CVT ratio.

2.0 BACKGROUND OF CVT

The basic of the CVT is actually similar to a variator which consists of a primary pulley, a secondary

pulley and a metal belt connecting these two pulleys. The variator geometry schematic diagram is given in

Figure 1. By assuming that the belt has a fixed length, does not slip and moves at perfect circles with

primary and secondary running radii Rp andRs, respectively, the tangential velocities of both pulleys andbelt will be the same.

Figure 1: Variator Geometry

The relationship between speed and running radii of the variator can be given as follows:

ppss RR = (1)

p

s

sr

= (2)

p

sCVT

R

Rr = (3)

The implicit relationship between belt lengthL and running radii can be defined as:

L= (+2 ) RP+ (-2 ) RS+2c Cos ( ) (4)

RP=RS+c sin ( ) (5)

The relationship between running radii and pulley position can be given as:

RP=Rp0 + (xP /tan ()) (6)

RS=Rs0 + (xS /tan ()) (7)

By solving equation (4) and (5), for c = 165 mm, and L= 645.68 mm, the relationship between running

radii and belt wrapped angle can be plotted as is illustrated in Figure 2.


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Figure 2: Relationship between running radii and belt wrapped angle.

By using equation (3) and data from Figure 2, the relationship between running radii and CVT ratiocan beplotted as shown in Figure 3.

Figure 3: Relationship between running radii and CVT ratio.

3.0 EMDAP CVT

The DC motor shaft of the EMDAP CVT is directly connected to a set of gear reducers and power screw

mechanism to axially move the pulley sheaves. A Van Doornes metal pushing V-belt is placed between

pulley sheaves, and runs on the surfaces of the sheaves. This metal belt connects the primary and

secondary pulleys to transmit the power and torque from the primary (input) to the secondary (output)

shaft by means of friction between belt and pulley sheaves contacts [9] and [10]. Both primary and

secondary running radii determine the CVT ratio, which is indirectly measured via the pulley position

sensors.

Both of the primary and secondary actuation system of the EMDAP CVT consists of a dc motor, two gearreducers, power screw mechanism and two movable metal pulley sheaves for clamping the metal belt. A

spring disc is placed at the back of each secondary pulley sheave to keep the belt taut and to reduce

excessive slip during ratio change.

The two gear reducers employed in the EMDAP-CVT system have an overall ratio of about 128:1. The

gear reducer input is coupled to the DC motor shaft, whereas its output is connected to the power screw

mechanism to move the pulley sheaves. The power screw mechanism converts every one rotational screw

movement to about 2-millimeter axial movement. The block diagram of the EMDAP-CVT system can be

seen in Figure 4.


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Figure 4: Block diagram of EMDAP-CVT

3.1. Proposed Controller

This paper proposes PD controllers for both of the DC motor systems to track the reference CVT ratio.

Based on Figure 3, the reference CVT ratio indirectly can be represented as a combination of its respective

reference primary running radius and reference secondary running radius. To satisfy the control objective,the primary DC motor system tracks the reference primary running radius, whereas the secondary DC

motor tracks the reference secondary running radius. The actual CVT ratio is calculated using actualvalues of primary and secondary running radii. The block diagram of the proposed controller is given in

Figure 5.

TR_set-to-Rp_set

Kd

Kp

d/dtPrimary

DC motorGear

ReducerTR_set

TR_set-to-

Rs_setKd

Kp

d/dt Secondary

DC motor

Gear

Reducer

Rp_act

Rs_act

+

-

+

++

- -

-

rg

+

+

TR_act

Figure 5: Block diagram of the proposed controller.

3.2. Tuning Parameters of PD Controller

The transfer function of a PD controller has the following form:

GPD = Kp+ Kds (8)

Another form of PD transfer function can be represented as:

GPD = Kp(1+ Td s) (9)

The value ofKd is described as :

Kd= Kp Td (10)

The Astrom-Hagglund method [11] is used to determine initial gains of PD controller, namely, critical

waveform oscillation period Tc and critical gain Kc. These two values can be obtained from a relay


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feedback experiment, which is a closed loop relay experiment involving a DC motor system as its plant,and a relay as its feedback controller. The block diagram for EMDAP CVT relay feedback experiment is

shown in Fiure 6.

TR_set-to-Rp_set

PrimaryDC motor

GearReducer

TR_set

Rp_act

+

- -

(a)

TR_set-to-

Rs_set

Secondary

DC moto r

Gear

Reducer

Rs_act

+

-

TR_set

(b)

Figure 6: Relay feedback controller for (a) primary and (b) secondary DC motor systems.

The critical gain resulted from relay feedback experiment can be defined as:

a

dKc

4= (11)

By using both values of Tc and Kc, the PD parameters can now be calculated using Ziegler-Nicholsformula [8]:

Kp = 0.6 Kc (12)

Td= 0.125 Tc (13)

4. RESULTS AND DISCUSSIONS

4.1. Experimental Setup

The EMDAP CVT experimental test rig is shown in Figure 7. The PD controller is implemented on this

EMDAP CVT system. The test rig consists of EMDAP-CVT gear box, AC Motor unit, interfacing unit,

Data Acquisition System (DAS) Card, Personal Computer (PC), and Power Supply unit. The schematicrepresentation of the test rig is given in Figure 8.


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Figure 7. Experimental Test rig.

EMDAP CVT gearbox has two DC motor systems for adjusting the pulley sheaves running radii, two

position sensors for measuring pulley positions and two encoders for measuring shaft angular speeds. The

AC motor unit acts as an engine, which drives the input shaft of the EMDAP CVT with a maximum speed

of 80 rpm. The interfacing unit allows the DAS card to read sensors and to control the DC motors. The

DAS card is fully controlled by computer via Matlab/Simulink Real Time Workshop package, which also

performs the controller task for the EMDAP-CVT. The sampling time for this controller is set to 0.1

seconds. Control performance of the PD controller is determined based on the overshoot, steady state error

and trajectory errors.

PRIMARY PULLEY

POSITION SENSOR

SECONDARY RPMSENSOR

DA TA

ACQUISITIONC A R D

D C M O T O R

D R I V E RPRIMARY DC

MOTOR

COMP UTE RDESIRED

TRANSMISSION

RATIO

Vinput

RU N/STOP

FORWARD/REVERSE

Vinput

RU N/ST OP

FORWARD/REVERSE

EMDAPCVTSECONDARY

PULLEY POSITIONSENSOR

SECONDARY

DC MOTOR

D C M O T O R

D R IV E R

PRIMARY RPMSENSOR

Figure 8: Schematic representation of test rig

4.2. Initial Parameters of PD Controllers

The EMDAP CVT ratio range is from 2.0 (underdrive) to 0.7 (overdrive). A step input corresponding to

CVT ratio of 1.35 is used as input for relay feedback experiment. The amplitude of the relay controller is


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set to 10, since the input voltage in the range of -5 to +5 volts is needed to drive the DC motor system.Figure 9 shows the result of relay feedback experiment for primary DC motor system. The Vpd acts as

reference input for primary pulley position, the Vpa represents the actual primary pulley position and Vrel is

the relay controller output.

Figure 10 shows the result of relay feedback experiment for secondary DC motor system. The Vsd acts as

reference input for secondary pulley position, the Vsa represents the actual secondary pulley position andVrel is the relay controller output.

Figure 9: Result of relay feedback controller for primary DC motor system.

Figure 10: Result of relay feedback controller for secondary DC motor system.

From Figure 9, the values ofTc is 8.167 s, a and dare 0.423 V and 10 V, respectively. By using equation(11), (12),(13), and (10), the Kpand Kd of the primary PD controller are 18.07 and 18.45, respectively.

Similarly, for Figure 10, the values of Tc is 5.66 s, a is 0.615 V, and dis 10 V. The Kpand Kd of thesecondary PD controller are 12.42 and 8.79 respectively.

4.3. Fine Tuning of PD Controller

The initial parameter values obtained from relay feedback experiment needs to be fine tuned. The tuning

process is conducted by examining the output responses of primary and secondary pulley position sensor,

VPPS and VSPS, respectively, when the ratio reference is up-shifted from 2 (underdrive) to 0.7 (overdrive)

and down-shifted from 0.7 (overdrive) to 2 (underdrive). Pulley position sensors detect the current pulleypositions ofxP and xS. By applying equation (6) and (7), the current running radii ofRp, and Rs can becalculated. Finally, by applying equation (3) the current CVT ratio can be determined.

The tuning process will only fine tune the proportional part of PD controller manually and leave thedifferential part unchanged. The output responses of the CVT ratio during fine tuning process for both

primary and secondary PD controller are shown in Figures 11 and 12, respectively.


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(a)

(b)

Figure 11: Response of the primary DC motor system: (a) overdrive (b) underdrive.

(a)

(b)

Figure 12: Response of the secondary DC motor system: (a) overdrive (b) underdrive.


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The experimentally fine tuned values of proportional parts of PD controllers derived from Figures 11 and12 are given in Table 1.

Table 1. Fine tuned proportional gains

CVT RatioPrimary DC

Motor System

Secondary DC

Motor SystemDecreasing Kp=8 Kp=4

Increasing Kp=12 K p=5

4.4. PD Controller Performance

The fine tuned proportional gains and unchanged differential gains are selected as final gains for PD

controllers of EMDAP CVT. The performances of these PD controllers are tested using square wave andsinusoidal input excitations. The results are given in Fig. 13 and Fig. 14.

(a)

(b)

(c)


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(d)

(e)

Figure 13: (a) CVT ratio response (b) Underdrive settling time (c) Overdrive settling time (d)

Underdrive steady state error. (e) Overdrive steady state error.

(a)

(b)

Figure 14: (a) CVT ratio response (b) CVT ratio tracking error.

Figure 13 shows the responses of CVT ratio output during application of a square wave excitation input.

It can be seen that time taken by the PD controllers to go from CVT ratio of 0.7 to 2 is about 13 seconds,

whereas, the time taken to go from CVT ratio of 2 to 0.7 is about 15 seconds. In addition, PD controller


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has successfully eliminated the overshoot and minimized the steady state error in the ranges of 0.0045 to0.003 (underdrive), and 0.0013 to 0.0034 (overdrive).

Figure 14 shows the responses of CVT ratio output during application of a sinusoidal wave excitation

input. It can be seen that the PD controller has successfully tracked the input trajectory with an error in

the range of -0.07 to 0.05.

5. CONCLUSION

In this research, PD controller was examined to reduce the overshoot, steady state error and trajectory

error for ratio control of EMDAP CVT system. The results of this work show that the application of

Astrom-Hagglund method and Ziegler-Nichols formula is capable of providing a practical solution for

obtaining initial parameters of the PD controllers. The application of experimentally fine tuning has

shown a good performance for the PD controller. However, the authors believe that this is just a first start,

with further works on PD auto tuning controller, better results can be obtained.

NOMENCLATURE

Rp, primary running radius

Rs secondary running radius

p angular speed of the primary shaft (input)

s angular speed of the secondary shaft (output)

,rs speed ratio

rCVT CVT ratio

L belt lengthc pulley center distance

half the increase in the wrapped angle on the primary pulley.

Rp0 minimum primary running radius,

Rs0 minimum secondary running radius, pulley wedge angle (11),xP primary pulley position, and

xS secondary pulley position.

Kp proportional gain

Kd derivative gainTd derivative time constant

Tc critical waveform oscillation period

Kc critical gain

d amplitude of the relay outputa amplitude of the waveform oscillation

Vpd input voltage for reference primary pulley position

Vpa input voltage for actual primary pulley positionVrel relay controller output voltageVsd input voltage for reference secondary pulley position

Vsa input voltage for actual secondary pulley position


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ACKNOWLEDGMENT

This paper was supported by E-SCIENCE Fund (2006-2008) Vot Number 79054.

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Jurnal Mekanikal 2008 BS KBT-2