Modeling and Controller Design for Non-Minimum Phase System with Application to XY-Table

Michael Jackson Patrick#1, Norlela Ishak#2 Mohd Hezri Fazalul Rahiman#3, Mazidah Tajjudin#4, Ramli Adnan#5, #Faculty of Electrical Engineering, UiTM Malaysia, 40450 Shah Alam, Selangor, Malaysia

1mj_29jack@yahoo.com.sg 2norlelaishak@salam.uitm.edu.my

3hezrif@ieee.org 4mazidah@salam.uitm.edu.my 5ramli324@salam.uitm.edu.my

Abstract— In discrete-time control system, model obtained from small or reducing sampling-time would produce a non-minimum phase model. Feed-forward controller designed of this model using the inverse transfer function of the closed-loop system would have internal stability that usually not guaranteed. To overcome this stability problem, ZPETC that capable of cancelling all poles, cancellable-zeros, and phase error was proposed by Tomizuka. This paper presents model identification and controller design for non-minimum phase system using the zero phase error tracking control (ZPETC) method. The non-minimum phase system model was approximated using Matlab System Identification Toolbox from open-loop input-output experimental data using 80ms sampling time. This approximated model was used in simulation and real-time studies. The effectiveness of ZPETC to overcome the non-minimum phase problem on XY-table shows that the proposed method gave acceptable performances. Keywords— System Identification, Non-minimum Phase System, Zero Phase Error Tracking Control, Real-time control.

I. INTRODUCTION XY-table has been widely used and of important element of

many Computer Numeric Controlled (CNC) processing facilities, e.g. the work feeder of CNC lathe, CNC milling and drill press, the worktable of laser processing, welding, dispenser, bonding, packing, drilling, laser cutting and painting. In general, XY-table is composed of X-axis and Y-axis motion mechanism where each motion axis is driven by an individual actuator system such as DC servo-motor through high-precision ball-screw.

Precision tracking control is important in a variety of modern CNC machines. It has been widely studied in [1-7]. The achievable precision tracking control is not only determined by the mechanical properties of the systems but strongly depends on modeling and controller design by utilizing good control strategies [1-4]. With the availability of modern tools such as Matlab System Identification Toolbox, model identification technique to approximate the plant model from open-loop input-output experimental data is easy to be implemented [1].

In discrete-time control system, model obtained from small or reducing sampling-time would produce a non-minimum phase model [8]. Feed-forward controller designed of this model using the inverse transfer function of the closed-

loop system would have internal stability that usually not guaranteed [3]. Therefore, in order to improve internal stability and performance of the system effectively in machining process, many studies were presented in [1-9].

The best solution is by using zero phase error tracking control (ZPETC) method as proposed by Tomizuka [3] as the feed-forward controller. This method successfully cancelled all poles, cancellable zeros, and phase error by introducing additional factors. The effectiveness of ZPETC to overcome the non-minimum phase zero problems on XY-table are discusses in this paper.

This paper was organized in the following manner: Section II describes the plant used; Section III describes model identification technique; Section IV describes the ZPETC design; Section V is on result and discussion; and finally Section VI is the conclusions.

II. PLANT The plant used in these studies is an XY-table (Googol

Tech GXY3030VD4 series) that is shown in Fig. 1. The name of XY-table is given because of the prime activity of X axis controls the cross-motion and Y axis controls toward or away from column. The plant worktable has a maximum travel of 60 cm (or 30 cm in both directions) in each axis.

Fig. 1 XY-table (Googol Tech GXY3030VD4 series)

This plant composed of GT-400-SV series motion controller, DC servo-motor, servo driver, electrical control box, motion platform and rotary encoder. In the system, there

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is a rotary incremental encoder (FXA-Z/6G05L2000BM) that attached to each axis of servo motor and is feed-backed as position signal with 2000 pulse/revolution. The mechanical part as the load of XY-table is a two axis servo motor where each axis is coupled with high-precision ball-screws, through a bracket and guided by sliding rod mechanism. This mechanical part drives the worktable to desire position.

III. MODEL IDENTIFICATION The signal given in Fig. 2 was used as an input signal to the

plant for model identification. The signal was generated using three different frequencies based on Eq. (1) and represented by Eq. (2)

�=

=P

isii ktaku

1cos)( ω (1)

ktktktkV sssin 5.0coscos610cos)( ++= (2)

(sec) : : :

timesamplingtfrequencyamplitudea

s

i

i

ω

From Eq. (2), when using three different frequencies for input signal, the models that can be obtained are limited to second and third-order only. Higher-order models may produce unstable output. In these studies, the third order ARX331 model was selected to represent the nearest model of true plant.

Fig. 2 Input signal for model identification

A. X-axis Model Identification The output signal of X-axis plant that was obtained using

input signal of Fig. 2 was sampled at 80ms and is given in Fig. 3. The input and output signals of Fig. 2 and Fig. 3 have to be divided into two parts, i.e. (1 to 150) samples and (151 to 300) samples. The first part of the input-output signals will be used to obtain the plant model and the second part of the input-output signal will be used to validate the obtained model.

Fig. 3 Output signal of X-axis plant using 80 ms sampling time and input signal of Fig. 2

Using Matlab System Identification Toolbox, the first part of the input-output signal produces the X-axis plant model, ARX331 in the form of discrete-time open-loop transfer function as given in Eq. (3)

321

321

1o

1o

0.1933z0.5451z0.6479z10.05178z0.1217z0.03698z

)(zA)(zB

−−−

−−−

−

−

+−−++= (3)

When Eq. (3) is simplified,

321

21-1

1o

1o

0.1933z0.5451z0.6479z1)1.4002z3.2910z(10.03698z

)(zA)(zB

−−−

−−

−

−

+−−++= (4)

with time delay, d=1. From Eq. (4), the zero polynomials obtained is

211c 1.4002z3.2910z1)(zB −−− ++= (5)

When Eq. (5) is factorized, the locations of zero are at 5021.0−=z , and 7889.2−=z . This means that the X-axis plant model obtained is a non-minimum phase model with one of the zeros situated far away from the unit circle. The pole-zero plot of Eq. (3) is given in Fig. 4.

Fig. 4 Pole-zero plot of Eq. (3)

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The second part of input-output signals was used to validate the obtained model of Eq. (3). The second part of the input signal was used as an input to the model and the output from the model will be compared with the second part of the output signal. The result can be seen from Fig. 5.

Fig. 5 Comparison between the model and plant output signal

Using the model selection criterion, the following information was obtained: Best Fit: 90.16%, Loss Function: 0.0051, and Akaike’s Final Prediction Error, FPE: 0.0054 Based on the smallest values criteria of FPE and best fit of 90.16%, this X-axis plant model can be accepted.

B. Y-axis Model Identification

The output signal of Y-axis plant that was obtained using input signal of Fig. 2 and sampled at 80 ms is given in Fig. 6. The input and output signals of Fig. 2 and Fig. 6 have to be divided into two parts, i.e. (1 to 150) samples and (151 to 300) samples.

Fig. 6 Output signal of Y-axis plant using 80 ms sampling time and input signal of Fig. 2

Using Matlab System Identification Toolbox, the first part of the input-output signal produces the Y-axis plant model,

ARX331 in the form of discrete-time open-loop transfer function as follows:

321

321

1o

1o

0.2056z0.4557z0.7495z10.03507z0.1054z0.03978z

)(zA)(zB

−−−

−−−

−

−

+−−++= (6)

When Eq. (6) is simplified,

321

21-1

1o

1o

0.2056z0.4557z0.7495z1)0.8816zz6496.2(10.03978z

)(zA)(zB

−−−

−−

−

−

+−−++= (7)

with time delay, d=1. From Eq. (7), the zero polynomials obtained is,

211c 0.8816zz6496.21)(zB −−− ++= (8)

When Eq. (8) is factorized, the locations of zero are at 3902.0−=z , and 2594.2−=z . This means that the Y-axis plant model obtained is a non-minimum phase model with one of the zeros situated far away from the unit circle. The pole-zero plot of Eq. (6) is given in Fig. 7.

Fig. 7 Pole-zero plot of Eq. (6)

The second part of input-output signals was used to validate the obtained model of Eq. (6). The result can be seen from Fig. 8.

Fig. 8 Comparison between the model and plant output signal

Using the model selection criterion, the following information was obtained:

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Best Fit: 85.76%, Loss Function: 0.0103, and Akaike’s Final Prediction Error, FPE: 0.0109 Based on the smallest values criteria of FPE and best fit of 85.76%, this Y-axis plant model can be accepted.

IV. CONTROLLER DESIGN A tracking control system with two-degree-of-freedom

controller (2-DOF) is shown in Fig. 9. In tracking control system without feed-forward controller, the reference signal continuously varying and mixed with the closed-loop system dynamics, which make tracking error always remain [3]. Note that the main function of feedback controller is regulation against disturbance input. The feed-forward controller is required such that the reference signal can be pre-shaped by the feed-forward controller, so that more emphasis to the frequency components that were not properly taken care-off by the feedback system can be provided [4].

Fig. 9 2-DOF Controller

The closed-loop system transfer function, clG without feed-forward controller can be given in terms of discrete-time model:

)()(

)( 1

11

−

−−− =

zAzBz

zGc

cd

cl (9)

Where dz − represent a d-step delay or sampling-time decrease caused by the delay in the plant and computation. The factor

)( 1−zBc can be factorized into two parts as in Eq. (10)

)()()( 111 −−−+− = zBzBzB ccc (10)

Where )( 1−+ zBc denotes the minimum phase factor that

includes cancellable zeros, and )( 1−− zBc denotes the non-minimum phase factor that includes non-cancellable zeros.

A. Zero Phase Error Tracking Control (ZPETC) The ZPETC as feed-forward controller is based on pole-zero

and phase cancellation reported in literature [1-9] can be expressed as in Eq. (11)

[ ] )()1(

)()()(

12

11

−+−

−−− =

zBB

zAzBzzG

cc

ccd

ff (11)

Where, dz represent a d-step ahead desired output. Thus, ZPETC method utilizes the future desired output in order to compensate for the d-step delay in Eq. (9) of closed-loop transfer function clG . The factor 2)]1([ −

cB is a scaling factor

which normalizing the low frequency gain of the overall transfer function between desired outputs to actual output toward unity. As a result, only gain error that caused by non-minimum phase zero remain. The ZPETC can be divided into three blocks as given in Fig. 10.

Fig. 10 ZPETC feed-forward controller

B. Simulation Studies Due to the effect of poles cancellation to the ZPETC

structure of Fig. 11, the control structure can be simplified as given in Fig. 12. From Fig. 12, the implementation of tracking control by simulation does not require the whole plant model transfer function. What was needed here is only the non-minimum phase factor of the plant model.

Fig. 11 Trajectory-ZPETC scheme

Fig. 12 Tracking control structure for simulation studies

C. Real-Time Studies The implementation of the proposed real-time tracking

control was based on the control structure given in Fig. 13. This control structure consists of two parts that are feedback control and feed-forward control. A pole-placement method was used for the design of feedback controller. This method enables all poles of the closed-loop system to be placed at desired location and providing satisfactory and stable output performance.

Fig. 13 Tracking control structure for real-time studies

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V. RESULT AND DISCUSSION This section presents the results of simulation and real-time

studies of a tracking control system using ZPETC which was implemented to XY-table. The simulation and real-time control were done using Matlab Simulink.

A. Single Axis XY-Table Trajectory Control The tracking input-output signals of the simulated and real-

time control are given in Fig. 14 and 15. The tracking input shape is chosen such that to demonstrate the ability of each axis controller to track the trajectory with changing frequency components. For X-axis tracking, the simulated and real-time control results are given in Fig. 14. The tracking performances by simulation in terms of roots mean square error (rmse) yield 0.0350mm and slightly higher by experiment which yield rmse of 0.1451mm. For Y-axis tracking, the simulated and real-time control results are given in Fig. 15. The tracking performances by simulation yield rmse of 0.0680mm and rmse of 0.1877mm by experiment. Smaller tracking error cannot be achieved by simulation and experiment for both X-axis and Y-axis due to the inability of ZPETC to approximate the overall gain transfer function of unity for higher frequency or at high speed condition. The plant-model mismatches also contributing higher tracking error.

Fig.14 Simulation and experimental results for x-axis movement to track a

given trajectory sinusoidal input with multiple frequencies

Fig.15 Simulation and experimental results for y-axis movement to track a

given trajectory sinusoidal input with multiple frequencies

B. Two Axis XY-Table Trajectories ZPETC The simulation and experimental results of plotting circular

shape of radius 20mm at frequency of 0.1 rad/s are given in Fig. 16 – Fig. 18. As we can observed, the tracking of XY motor following the desired trajectory inputs could hardly be seen the differences. The simulation results of Fig. 16 show single axis movement for circular plot with tracking errors, rmse of 0.0197mm for X-axis and 0.0392mm for Y-axis. The experimental results of Fig. 17 show single axis movement for circular plot with tracking errors, rmse of 0.1022mm for X-axis and 0.1217mm for Y-axis. The experimental results of Fig. 18 show 2D movement for circular plot with good tracking capabilities.

Fig.16 Simulation result of single X-Y movement for circular plot

Fig.17 Experimental result of single X-Y movement for circular plot

Fig.18 Experimental result of 2D movement for circular plot

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The simulation and experimental results of plotting lemniscates shape are given in Fig. 19 – Fig. 21 using two difference references. As we can observed, the tracking of XY motor following the desired trajectory inputs could hardly be seen the differences. The simulation results of Fig. 19 show single axis movement for lemniscates plot with tracking errors, rmse of 0.0099mm for X-axis and 0.04131mm for Y-axis. The experimental results of Fig. 20 show single axis movement for lemniscates plot with tracking errors, rmse of 0.0475mm for X-axis and 0.0913mm for Y-axis. The experimental results of Fig. 21 show 2D movement for lemniscates plot with good tracking capabilities.

Fig.19 Simulation result of single X-Y movement for lemniscates plot

Fig.20 Experimental result of single X-Y movement for lemniscates plot

Fig.21 Experimental result of 2D movement for lemniscates plot

VI. CONCLUSION

The studies on plant modeling and controller design using zero phase error tracking controller for non-minimum phase system with application to XY-table were presented. Model identification using Matlab System Identification Toolbox is very much simpler to be implemented as compared to deriving mathematic model using physic laws. ZPETC gave satisfactory performance and almost perfect tracking for a given input for both X and Y-axis tracking but, the gain error still present. The tracking performances from the simulation and experiment show that ZPETC can be used in precision machining.

ACKNOWLEDGMENT The authors would like to thanks and acknowledge the

FRGS-RMI-UiTM for financial support of this research work.

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