Position Control of a Large Antenna System

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Poition Control of a Large Antenna Sytem uldip S. Rattan Department of Electrical Engineering Wright State Univerity Dayton, OH 45435 krattan@c.wright.

1 Poition Control of a Large Antenna Sytem uldip S. Rattan Department of Electrical Engineering Wright State Univerity Dayton, OH krattan@c.wright.edu ABSTRACT Thi report decribe the deign of a poition control ytem for a large antenna ytem. The antenna i ued to receive atellite ignal and mut accurately track the atellite a it move acro the ky. The objective i to deign a Proportional-plu-Integral-plu-Derivative controller to atify certain performance pecification. The reult of the deign how that all the deired performance pecification are atified. I. Introduction The block diagram of the poition control ytem for a large antenna i hown in Figure. The objective i to deign a continuou controller, if neceary, to atify the following performance pecification. The teady-tate error to a ramp input %. Maximum percentage overhoot to a tep input 5% or the damping ratio, ζ Settling time (% criterion) < econd or the natural frequency, ω n >.5 rad/ Figure. Block diagram of the poition control ytem for a large antenna. Figure. A large antenna ytem.

2 The large antenna, hown in Figure, i ued to receive atellite ignal and mut accurately track the atellite a it move acro the ky. The control ytem ue an armature controlled motor and the tranfer function of the motor and antenna i given by G p ( ) ( + 5)( + ) It i alo required to evaluate the effect of diturbance, D() Q/, on the output Y(). The analyi of the uncompenated ytem i carried out in Section III. A jutification of the type of controller needed i alo given in Section III. The deign of the PID controller i carried out in Section IV and the reult are preented in Section V. Finally, the concluion are drawn in Section VI. III. Analyi of the Uncompenated Sytem The pecification that the teady-tate error of the ytem to a unit ramp input mut be le than or equal to. require the velocity error contant mut be greater than or equal to. The velocity error contant of the uncompenated ytem (G c () ) i given by v lim G ( ) p 5 Therefore, a gain of 5 mut be added to the ytem to atify the teady-tate error pecification. However, from a practical conideration, thi gain i too large, and the uncompenated ytem with a gain of 5 i untable with a phae margin of approximately -4 degree. If a phae-lead type of a controller i to be deigned for the ytem with a gain of 5, the controller will have to provide a phae of more than degree. Thi i not poible for a proportional plu derivative (PD) controller or a ingle-tage phaelead controller. Thu, to atify the teady-tate error pecification without exceive gain, we need to increae the type of the ytem by one. When a type one ytem i converted to a type- ytem by a Proportional-plu-Integral (PI) controller, the gain no longer affect the teady-tate error, and i alway zero for a table ytem with a ramp input. The root locu of the uncompenated ytem i hown in Figure 3. It can be een from thi figure that for maller value of gain, the value of natural frequency (ω n ) are mall and the ζ value are large, however, for larger value of, the ω n value are large and the value of ζ are mall. For example, for 4.8, the value of ζ i and ω n.6 rad/, however, for, the value of ω n 4. rad/ and ζ.35. Alo, the PI controller puhe the root locu to the right, and hence the overhoot and ettling time pecification can not be met by a PI controller. PI controller can be deigned to atify the ettling time pecification, and then a Proportional-plu-Derivative (PD) controller can be cacaded with the PI controller to add damping to the ytem. Thi lead to the motivation of uing a Proportional-plu-Integral-plu Derivative (PID) controller, o that the bet feature of each of the PI and PD controller are utilized. The tranfer function of a PID controller can be written a I Gc ( ) P + D + () The PID controller given in equation () conit of a PI portion connected in cacade with a PD portion and can be written a in equation ()

3 Root Locu Imag Axi Real Axi Figure 3. Root locu of the uncompenated ytem. I Gc ( ) ( + D )( + ) () where P D I D I + D I (3) The proportional contant of the PD portion i et to unity, ince we need only three parameter in the PID controller. In the deign of the PID controller, the PI portion i deigned to take care of the ettling time and the damping of the ytem i taken care of by the PD portion or the PD portion can be deigned firt by canceling one of the pole, (which take care of the ettling time), and then the PI portion can be deigned to take care of the damping. In thi paper, the PD portion i deigned by canceling one of the pole and then the PI controller i deigned to atify the overhoot pecification. IV. Deign of PID Controller In thi report, a PID controller i deigned by firt conidering that the PD portion i only in effect. The value of D i elected by canceling the pole at -5. Thi puhe the root locu to the left and ha the effect of increaing the natural frequency of the ytem. Finally, the PI portion i deigned by electing the appropriate value of and I to atify the overhoot requirement.

4 4. Deign of the PD Portion of the Controller The tranfer function of the PD portion of the PID controller i given by G ) ( + ) (4) c( D The forward-path tranfer function of the ytem compenated by the PD controller i then given by G( ) ( + D ) ( + 5)( + ) (5) The pole-zero cancellation method i ued to deign the PD portion of the controller. Canceling the pole at -5 with the zero provided by the PD controller ( D.) ha the effect of puhing the root locu to the left and in the proce increae the natural frequency of the ytem a hown in Figure 4. The forward path tranfer function of the ytem with D. i given by G ( ) (6) ( + ) It can be een from Figure 4 that for a gain of.5, the natural frequency i 5 rad/ and the damping ratio i. Thi gain i obtained from the PI portion of the controller. Root Locu Imag Axi Frequency 5 rad/ Damping ratio Real Axi Figure 4. Root locu of the cloed-loop ytem compenated with a PD controller.

5 4. Deign of the PI Portion of the Controller The tranfer function of the PI portion of the PID controller can be written a G c I ( + ) ( ) (7) The forward-path tranfer function of the ytem compenated by the PID controller become G ) ( ( + ( + ) I ) (8) Following the guideline of chooing a relatively mall value of I /, we let I /.. The root locu of the ytem with the zero fixed at -. and varying the value of i hown in Figure 5. It can be een from thi figure that.5 atifie the ettling time and the overhoot pecification. Root Locu Imag Axi p Sytem: y Gain: Pole: i Damping:.5 Overhoot (%): Frequency (rad/ec): p -.5 p p Real Axi. Root Locu Imag Axi -. p p Real Axi Figure 5. Root locu of the ytem compenated by a PID controller.

6 The tranfer function of the PID controller i thu given by where P.55, D.5 and I.5. V. Reult.5 ( +.) ( +.).5 ( + 5) ( +.) G c ( ) The cloed-loop tranfer function of the ytem compenated by the PID controller deigned in Section 4 i given by Y ( ) R( ) 3 5 ( +.) (9) The unit tep repone of the cloed-loop ytem i hown in Figure 6. It can be een from thi figure that that all the performance pecification are atified. The bode plot of the compenated ytem hown in Figure 7 how that the G.M. i and the P.M. i 86 o. Step Repone.8 Settling Time.3 ec No overhoot Amplitude Time (ec) Figure 6. Unit tep repone of the cloed-loop compenated ytem. The cloed-loop tranfer function of the ytem with the output, Y(), and input D() i given by Y ( ) D( )

7 The repone of the ytem to a unit tep diturbance i hown in Figure 8. It can be een from thi figure that the ytem i very robut, i.e., the repone to diturbance i very mall. Bode Diagram Gm Inf, Pm deg (at.9 rad/ec) Phae (deg) Magnitude (db) Frequency (rad/ec) Figure 7. Bode plot of the compenated ytem. x Step Repone 3.5 Amplitude Time (ec) Figure 8. Repone of the ytem due to unit diturbance.

8 VI. Concluion Deign of an antenna poition control ytem i preented in thi report. The large antenna i ued to receive atellite ignal and mut accurately track the atellite a it move acro the ky. The objective i to deign a Proportional-plu-Integral-plu-Derivative controller to atify the deired performance pecification. The deign of the PID controller i carried out by firt deigning a PD controller to atify the ettling time requirement and then a PI controller i deigned to atify the overhoot requirement. The reult of the deign how that the teady-tate error of the compenated ytem to a ramp input i le than %, the ettling time i.3 econd and the cloed-loop ytem ha no overhoot. Therefore, all the deired performance pecification are atified. Reference. Benjamin C. uo and Farid Golnaraghi, Automatic Control Sytem, 8 th Edition, John Wiley & Son, Inc.. Richard C. Dorf, and Robert H. Bihop, Modern Control Sytem, 8 th Edition, Addion-Weley.

Chapter Introduction

Chapter Introduction Chapter-6 Performance Analyi of Cuk Converter uing Optimal Controller 6.1 Introduction In thi chapter two control trategie Proportional Integral controller and Linear Quadratic Regulator for a non-iolated