[ieee 2010 ieee control and system graduate research colloquium (icsgrc) - shah alam, malaysia...

Vibration Control of Two-Mass Rotary System Using Improved NCTF Controller for Positioning

Systems Mohd Fitri Mohd Yakub#1, Wahyudi Martono*2, and Rini Akmeliawati*3

#Department of Electrical Engineering UTM International Campus, Kuala Lumpur, Malaysia

[email protected]*Department of Mechatronic Engineering

International Islamic University Malaysia (IIUM), Kuala Lumpur, Malaysia [email protected]@iiu.edu.my

Abstract— In this paper, a nominal characteristic trajectory following (NCTF) controller for point-to-point (PTP) positioning system for two mass rotary system is introduced and its performance is evaluated. Generally, the NCTF controller consists of a nominal characteristic trajectory (NCT) and a compensator. The objective of the NCTF controller is to make the object motion follow the NCT and end at its origin. The NCTF controller is designed based on a simple open-loop experiment of the object. The parameters and an exact model of the plant are not necessary for controller design. This paper presents a method to improve the existing NCTF controller for two mass rotary positioning system by adding a notch filter as a compensator to eliminate the vibration due to the mechanical resonance. They can often remove resonance without compromising performance. The improved NCTF controller is evaluated and discussed based on results of simulation. The effect of the design parameters on the robustness of the NCTF controller to inertia and friction variations is evaluated and compared with conventional PID controller. It is shown that improved NCTF controller is better than conventional PID controller. Keywords— Vibration, improved NCTF, two-mass system, notch filter, PID controller

I. INTRODUCTION The problem on vibration suppression and disturbance

rejection in flexible system originates in steel rolling mill system, where the load is coupled to the driving motor by a long shaft. The small elasticity of the shaft is magnified and has a vibration effect on the load position. Positioning systems play an important role in industrial engineering applications such as advanced manufacturing systems, semiconductor manufacturing system and robot systems As the newly required fast response speed and high accuracy is very close to the first resonant frequency, only conventional P&I control are no longer effective. To overcome the problem, various control strategies have been proposed mainly for controlling the two-mass system, the simplest model [1].

A nominal characteristic trajectory following (NCTF) controller as practical controller for point-to-point positioning systems for one mass system had been proposed since 2002 [2]. The NCTF controller consists of two elements namely a nominal characteristic trajectory (NCT) and a compensator. It had been reported that the NCTF had a good positioning performance and robustness to parameters variations [2].

However, NCTF controller that has been proposed is designed based on one-mass rotary system. The positioning system can only be assumed as one-mass positioning system in the case a rigid coupling is used and there are no flexible elements in between motor and load. On the other hand, the systems should be modeled as multi-mass system when flexible couplings with low stiffness or other flexible elements are used to connect the actuator to other elements. Some application like in robot industry which have a long arm for linear system or long shaft in rotary system will be considered as two mass or multi mass systems. In addition to spring forces, compliant couplings also provide viscous damping forces. Damping forces are produced in proportion to the velocity difference between the motor and load, rather than position difference, as is the case with spring forces [3].

Therefore, the existing NCTF controller can not be used directly in the case there is a flexible connection between elements of the positioning systems. Improvements in the design of NCT and compensator are required to make the NCTF controller is suitable for two-mass positioning system.

In this paper, the improved NCTF controller is expected to control the position and reduces the vibration by using a notch filter that can often remove resonance without compromising performance. The performances of the improved NCTF controller is eavaluated and compared with the conventional PID controller.

II. MODEL OF THE SYSTEM

The schematic diagram of the two mass system is illustrated in Fig. 1. Two masses, having the moments of inertia Jm and Jl, are coupled by low stiffness shaft which has

2010 IEEE Control and System Graduate Research Colloquium

978-1-4244-7240-6/10/$26.00 ©2010 IEEE 61

the torsion stiffness Ks and a damping. Damping due to the friction is very small so it can be neglected without affecting analysis accuracy.

For the case that the system can be accurately modeled without considering the major nonlinear effects by the speed dependent friction, dead time and time delay, a linear modelfor two mass mechanical system can be obtain using the conventional torque balance rule [4]:

(1)

(2)

(3)

with

, ,

where Jm and Jl (kgm2) are the motor and load moments of inertia, wm and wl (rad/s) are the motor and load angular speed, Tm and Tl (Nm) are the motor and load disturbance torque, Ts(Nm) is the transmitted shaft torque, Bm and Bl (Nms/rad) are the viscous motor and load frictions, Bs (Nms/rad) is the inner damping coefficient of the shaft and Ks (NM/rad) is the shaft constant.

Fig. 1 Schematic diagram of two mass system.

III. NCTF CONTROL CONCEPT

The structure of NCTF control system is shown in Fig. 2. The NCTF controller consists of a NCT and a compensator. NCTF controller works under the following two assumptions[5]:a) A DC or an AC servo motor is used as an actuator of the object. b) PTP positioning systems are discussed, so �r is constant and �r’ = 0.

Fig. 2 Structure of NCTF control system for PTP positioning.

The objective of the NCTF controller is to make the object motion follow the NCT and end at the origin of the phase plane (e, e’). Signal up shown in Fig. 2, represents the difference between the actual error rate e’ and that of the NCT. The value of up is zero if the object motion perfectly follows the NCT. The compensator is used to control the object so that the value of up, which is used as an input to the compensator, is zero.

Fig. 3 shows an example of object motion controlled by the NCTF controller. The object motion comprises two phases: one is the reaching phase and the other, the following phase. In the reaching phase, the compensator forces the object motion to reach the NCT as fast as possible. In the following phase, the compensator controls the object motion to follow the NCT and end at the origin. The object motion stops at the origin, which represents the end of the positioning motion. Thus, the NCT governs the positioning response performance.

Error, e

Erro

r rat

e, e

' NCT

o

Object motion

RP: Reaching phaseFP: Following phase

FP

RP

Fig. 3 NCT and object motion.

The NCTF controller is designed based on a simple open-loop experiment of the object as follows [6]:1) Open-loop-drive the object with a stepwise input and measure the displacement and velocity responses of the object.2) Construct the NCT by using the object responses. Since the NCT is constructed based on the actual responses of the object, the NCT includes effects of nonlinear characteristics such as friction and saturation.3) Design the compensator by using the NCT information. The NCT includes information of the actual object parameters. Therefore, the compensator can be designed by using only the NCT information.

mmsmm

m BTTdt

dJ �

��

��

�

lllsl

l BTTdt

dJ ��

��

�

)()( lmslmss BKT ��

mm

dtd

��

� ll

dtd

��


62

Due to the fact that the NCT and the compensator are constructed from a simple open-loop experiment of the object, the exact model including the friction characteristic and the conscious identification task of the object parameters are not required to design the NCTF controller. The controller adjustment is easy and the aims of its control parameters are simple and clear.

IV. IMPROVED NCTF CONTROLLER DESIGN

A. NCT ConstructionsIn order to construct the NCT, a simple open-loop

experiment has to be conducted. In the experiment, an actuator of the object is driven with a stepwise input, the load displacement and load velocity responses of the object are measured. Fig. 4 shows the stepwise input, load velocity and load displacement responses of the object. In this case, the object vibrates due to its mechanical resonance. In order to eliminate the influence of the vibration on the NCT, the object response must be averaged. The parameter of the object used only for making simulation is shown in Table I.

0 1 2 3 4 5 6 7 8 9 10-80

-60

-40

-20

0

20

40

60

80

Time

Inpu

t, D

isp

lace

me

nt,

Ve

loci

ty stepwiseloaddisplacementloadvelocity

Fig. 4 Input and actual object response.

TABLE INOMINAL OBJECT PARAMETERS

Parameter Value UnitMotor inertia, Jm 17.16e-6 Kgm2

Inertia load, Jl 24.17e-6 Kgm2

Stiffness, Kc 0.039 Nm/radMotor resistance, R 5.5 �Motor inductance, L 0.85e-3 HTorque constant of the motor, Kt 0.041 Nm/AMotor voltage constant, Kb 0.041 Vs/radFrictional torque, Tf 0.0027 NmMotor viscous friction, Bm 8.35e-6 Nms/radLoad viscous friction, Bl 8.35e-6 Nms/rad

In Fig. 5, moving average filter is used to get the averaged response because of its simplicity [7]. The moving average filter operates by averaging a number of points from the object response to produces each point in the averaged response. Theaveraged velocity and displacement responses are used to determine the NCT. Since the main problem of the PTP motion control is to stop an object at a certain position, a deceleration process (curve in area A of Fig. 5) is used. Variable h in Fig. 5 is the maximum velocity, which depends

on the input step height. From the curve in area A and h in Fig.5 (a), the NCT in Fig. 5 (b) is determined.

There are two important parameters in the NCT as shown in Fig. 5(b): the maximum error rate indicated by h, and the inclination of the NCT near the origin indicated by m. As discussed in the following section, these parameters are related to the dynamic parameters of the object. Therefore, the parameters are used to design the compensator.

0 1 2 3 4 5 6 7 8 9 10-40

-20

0

20

40

60

80

Time

Inpu

t, D

isp

lace

ment,

Velo

city stepwise

averaged displacementaveraged velocity

U

h

A

A

h

r

a) Input and averaged object response

-1.5 -1 -0.5 0 0.5 1 1.5-200

-150

-100

-50

0

50

100

150

200

Error, e

Erro

r rat

e, e

'

A

hA

m

o

h

b) Nominal characteristic trajectory

Fig. 5 Construction of the NCT.

An exact modeling including friction and conscious identification processes are not required in the NCTF controller design. The compensator is derived from the parameter m and h of the NCT. Since the DC motor is used as the actuator, the simplified object can be presented as a following fourth order system:

2

2

2

2

2)()()(

)(fff

flo ssss

KsUs

sG��

��

��

�� (4)

where �l (s) represents the displacement of the object, U(s), the input to the actuator and K, �, �2 and �f are simplified object parameters. The NCT is determined based on the averaged object response which is does not include the vibration. So, it can be assumed that the averaged object response is a response to the stepwise inputs of the averaged object model as follows:

)()(

)(

2

2

��

�ssK

sUsav (5)


63

where �av(s) is the averaged load displacement, U(s), input to the actuator and K and �2 are simplified object parameters that related to the NCT information. The relations between �� 2 and the NCT information are[6]:

m��2� (6)

ru

hK �� (7)

B. Improved Compensator Design

The following PI and notch filter compensator is proposed for a two mass systems [8]:

��

��

��

��

)2()2()(

)( 2

2

ooo

fffdcipc ss

ssKs

KsKsG

��

(8) The PI compensator is adopted for its simplicity to forces the object motion to reach the NCT as fast as possible and control the object motion to follow the NCT and end at the origin. The PI compensator parameters Ki and Kp are designed by using �n and � as the design parameters as follows:

mh

uK

K rnnp

�� 22

2

�� (9)

mh

uK

K rnni

2

2

2 ��

�� (10)

A higher �n and a larger � are preferable in the compensator design. However, while choosing � and �n, the designer must consider the stability of the control system. In continuous system, a linear stability limit can be calculated independently of the actual mechanism characteristic. However, the stability limit is too limited because of neglected Coulomb friction is known to increase the stability of the system, allowing for the use of higher gains than those predicted by a linear analysis [9].

The higher gains are expected to produce a higher positioning performance. Thus the practical stability limit is necessary for selecting the higher gain in a design procedure. The selection of �n and � are chosen to have 40% of the values of �� practical, so that the margin safety of design is 60% [10]. During the design parameter selection, the designer may be tempted to use large values of �n and � in order to improve the performance. However, excessively large values of �n will cause the controller to behave as a pure integral controller, which may lead to instability. Therefore, the choise of �n should start with small values and progress to large ones and can not opposite.

In many systems, the mechanical couplings between the motor, load, and sensor are not perfectly rigid, but instead act like springs. Here, the motor response may overshoot or even oscillate at the resonance frequency resulting in longer settling time. The most effective way to deal with this torsional resonance is by using an anti resonance notch filter.

According to standard frequency analysis, resonance is characterized by a pair of poles in the complex frequency plane. The imaginary component indicates the resonant frequency, while the real component determines the damping level. The larger the magnitude of the real part, the greater the damping [11].

A notch filter consists of a pair of complex zeros and a pair of complex poles. The purpose of the complex zeros play by�f and �f is to cancel the resonance poles of the system. The complex poles which are �o and �o, on the other hand, create an additional resonance and to increase a stability of gain margin for the plant. If the magnitude of the real value of the poles is large enough, it will result in a well damped response. The ratio between �f and �o will determine how deep the notch in order to eliminate the resonant frequency of the plant. Parameter Kdc will be affected in steady state condition when the transfer function of the notch filter becomes one.

Fig. 6 shows where the poles and zeros of the system are located on the s plane. The poles marked A are the ones due to the mechanical resonance. These are cancelled by the complex zeros marked by B.

Although it is assumed that the notch filter completely cancels the resonance poles, perfect cancellation is not required. As long as the notch zeros are close enough to the original poles, they can adequately reduce their effect, thereby improving system response.

-160 -140 -120 -100 -80 -60 -40 -20 0 20

-300

-200

-100

0

100

200

300

Root Locus

Real Axis

Imag

inary

Axis

A

A

B

B

a) Poles and Zeros cancellation of the notch filter

-200

-150

-100

-50

0

50

Mag

nitu

de (

dB)

System: sysL1Frequency (rad/sec): 281Magnitude (dB): -76

100

101

102

103

104

-360

-270

-180

-90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

plantplant w ith nf

b) The ratio between �f and �o = 0.6


64

-200

-150

-100

-50

0

50

Mag

nitu

de (

dB)

100 101 102 103 104-360

-270

-180

-90

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Plant

Plant, wo=330

Plant, wo=1000

c) The effect of �o

Fig. 6 Bode plot response of the system

V. CONVENTIONAL PID CONTROLLERS

Conventional PID controllers were designed based on a Ziegler Nichols and Tyres Luyben closed loop method, using proportional control only. The proportional gain is increased until a sustained oscillation output occur which giving the sustained oscillation, Ku, and the oscillation period, Tu are recorded. The tuning parameter can be found in Table II [12].

TABLE IICONTROLLER TUNING RULE PARAMETERS

Controller Kp Ti TdZiegler NichlosTyres Luyben

Ku/1.7 Tu/2 Tu/8Ku/2.2 Tux2.2 Tu/6.3

VI. SIMULATION RESULTS

The significance of this research lies in the fact that a simple and easy controller can be designed for high precision positioning system which is very practical. By improving the NCTF controller, it will be more reliable and practical for realizing high precision positioning systems for two-mass positioning systems compared with conventional PID in term of controller performances.

According to Fig. 5, the inclination, m and maximum error rate, h of the NCT are 579.49 and 38.98 respectively. When designing the PI compensator, design parameters for � and �nare chosen as 10.5 and 6.2 in order to evaluate the performance of NCTF controller. It is compared with PID controller with tuned using Ziegler-Nichols and Tyres Luyben method. Table III shows the PID and PI compensator controller parameters.

TABLE IIICONTROLLER PARAMETERS

Controller Kp Ki KdNCTFZiegler Nichols PIDTyres Luyben PID

6.81e-2 1.98e-2 -3.348 42.65 0.06572.536 7.342 0.0632

A. Controller Performances

The performance is evaluated based on percentage overshoot, settling time, and steady state error on rotary two mass positioning system. Fig. 7 shows comparison simulated

step response due to 30 degree and 90 degree with normal object, increase inertia and friction object.

From Fig. 7, it shows that improved NCTF control system gives better positioning performance without overshoot for 30 deg step input and very small percentage overshoot, 6.4% while increasing the position input to 90 deg for normal plant. The overshoot will increase proportionally with the higher inertia object while the conventional PID becomes unstable for the same object. It also shows that the settling time for improved NCTF controller is much faster than conventional PID controller for various step inputs and higher inertia object. For steady state error, improved NCTF controller gives better response with smaller steady state error compared with the conventional PID controller.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-20

0

20

40

60

time, sec

Lo

ad

dis

pla

ce

me

nt,

30

de

g

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1

0

1

2

3

time, sec

Co

ntr

ol sig

na

l,V

Zeigler-NicholsTyres-LuybenNCTF


a) Comparison due to step response, Normal object

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50

0

50

100

150

200

time, sec

Lo

ad

dis

pla

ce

me

nt,

90

de

g

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

0

2

4

6

8

time, sec

Co

ntr

ol sig

na

l,V



b) Comparison due to step response, Increase inertia object (2xJl)


65

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

0

10

20

30

40

50

time, sec

Lo

ad

dis

pla

ce

me

nt,

30

de

g

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2

time, sec

Co

ntr

ol s

ign

al,

V



c) Comparison due to step response, Increase inertia object (5xJl)

Fig. 7 Performance comparison of the step response due to increase inertia.

Furthermore, Fig. 8 shows a comparison of step responses due to a 30 deg and 90 deg step input when the controllers are implemented on increased friction object (2xft) and (5x ft).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-20

0

20

40

60

time, sec

Load

dis

plac

emen

t,30

deg

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50

0

50

100

150

time, sec

Load

dis

plac

emen

t,90

deg


NCTFZeigler-NicholsTyres-Luyben

a) Comparison due to step response, Increase friction object (2xft)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

0

10

20

30

40

50

time, sec

Load

dis

plac

emen

t,30

deg

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-50

0

50

100

150

time, sec

Load

dis

plac

emen

t,90

deg



b) Comparison due to step response, Increase friction object (5xft)

Fig. 8 Performance comparison of the step response due to increase friction.

From Fig. 8, it shows that the improved NCTF control system has much better positioning performance as percentage overshoot is smaller and it’s settling time is the fastest for various step inputs and increased friction object compared to both conventional PID controllers. However, improved NCTF controller gives a lower steady state error compared to Ziegler Nichols method but larger steady state error than that of Tyres

Luyben method. All process within 10 second simulation time.

The error dynamics in term of the error, e and the error dot, e’ are shown in Fig. 9, which shows that the motions reach the NCT and then try to follow the trajectory as expected in NCTF control concept.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

0

10

20

30

40

time, sec

Dis

plac

emen

t,30

deg

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-20

-15

-10

-5

0

5

Error , rad

Err

or ra

te, r

ad/s

NCTPI

NCTF

Fig. 9 Object motions for a 30 deg step input.

VII. SUMMARY

The improvement of NCTF controller as a new practical control for two mass positioning systems has been introduced and discussed. The improved NCTF controller consists of the NCT and the PI with notch filter compensator. The NCT is constructed using the object response data in a simple open-loop experiment and the compensator parameters are designed based on the NCT. The effectiveness of the improved NCTF controller is examined by simulation and it showed that the improved NCTF controller is much more effective and robustness then the conventional PID controller for positioning systems. It also showed that vibration cause by mechanical resonance was effectively compensated by added a notch filter. The comparison performance for two mass systems based on experimental is left for further work.

ACKNOWLEDGMENT

This research is supported by Ministry of higher education IIUM fundamental research grant scheme (IFRG) no IFRG0702-60.

REFERENCES[1] Amstrong-Helouvry B, Dupont P. and De Witt C, A Survey of Models,

Analysis Tools and Compensation Method for the Control of Machines with Friction, Automatica, Vol. 30, pp. 1083-1138, 1994.

[2] Wahyudi, New Practical Control of PTP Positioning Systems, Ph.D Dissertation, Tokyo Institute of Technology Japan., 2002.

[3] How to work with mechanical resonance in motion control systems.Control Engineering, Vol. 47, No. 4, 2000, p. 5.

[4] Robert L.Woods and Kent L.Lawrence, Modelling and Simulation of Dynamic Systems, Prentice Hall Inc, 1997.

[5] Wahyudi, Sato K. And Shimokohbe A, Robustness Evaluation of New Practical Control Method for PTP Postioning Systems, Proceeding of 2001 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp 843-848, July 2001.

[6] Wahyudi and Albagul A (2004). Performance improvement of practical control method for positioning system in the presence of actuator saturation, Proceedings of 2004 IEEE International Conference on Control Applications. Taipei, 2-4 September, pp. 296-302.


66

[7] Oppenheim A.V. & Schafer R.W, Discrete Time Signal Processing.Englewood Cliffs, Prentice Hall, 1999.

[8] Fitri.MY, Wahyudi, Rini Akmeliawati & Andika A. Wijaya, NCTF Control Method of Two Mass System for PTP Positioning, Proceedings of the International Conference on Man-Machine Systems (ICoMMS), Penang, 11-13 October 2009, p.4A4 1 - 4A4 6

[9] Townsend WT and Kenneth Salisbury J, The effect of coulomb friction and stiction on force control, IEEE Int Conf Robot Automat 1987;4:883-9

[10] Guilherme Jorge Maeda & Kaiji Sato, Practical control method for ultra-precision positioning using a ballscrew mechanism. Precision Engineering Journal, Vol. 32(2008)309-318, November 2007

[11] William East & Brian Lantz, Notch Filter Design, August 29, 2005[12] Astrom K. & Hagglund T, PID controllers: Theory, design and tuning.

Instrument Society of America, 1995.[13] De Wit C., Olsson H., Astrom K.J & Lischinssky, Dynamic friction

models and control design, Proceedings American Control Conference, San Francisco, USA, 1993, pp. 1920-1926


67

[ieee 2010 ieee control and system graduate research colloquium (icsgrc) - shah alam, malaysia...

Documents