Cansever, G.

Gulez, Kayhan

Kucukdemiral, I. B.

English

This note considers the ways of obtaining the synthesis equations of the dynamic controller for a DC Servo machine where the closed-loop system poles will move in a -stable region, in spite of the changing system transfer function coefficients by the perturbations. First of all, the mathematical model of a dc servo machine isobtained. Depending upon the varying system transfer function, design equations of the dynamic compensator are found out. These equations includes the ways of choosing the controller coefficients KP, KI, KD which will give a desired response for the tracking system

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Synthesis inequalities for the robust PID controller for a DC Servo motor andImplementation by using digital signal processorKucukdemiral, I. B.; Cansever, G.; Gulez, KayhanPublished in:Proc. International Conference on Signal Processing Applications and Technology (ICSPAT'99), Orlando,Florida, USAPublication date:1999Document VersionPublisher's PDF, also known as Version of recordLink to publication in ResearchOnlineCitation for published version (Harvard):Kucukdemiral, IB, Cansever, G & Gulez, K 1999, Synthesis inequalities for the robust PID controller for a DCServo motor and Implementation by using digital signal processor. in Proc. International Conference on SignalProcessing Applications and Technology (ICSPAT'99), Orlando, Florida, USA. ICSPAT 99, InternationalConference on Signal Processing Applications and Technology, Orlando, FL., United States, 1/11/99.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.Take down policyIf you believe that this document breaches copyright please view our takedown policy at https://edshare.gcu.ac.uk/id/eprint/5179 for detailsof how to contact us.Download date: 02. Jan. 2022Synthesis Inequalities for the Robust PID Controller for a DC Servo Motor andImplementation by Using a Digital Signal Processor• brahim Beklan KÜÇÜKDEM• RAL, Kayhan GÜLEZ, Galip CANSEVERElectrical Engineering Department, Y• ld• z Technical UniversityY• ld• z/• stanbul  TURKEY  P.C: 80750Abstract  This note considers the ways of obtaining thesynthesis equations of the dynamic controllerfor a DC Servo machine where the closed-loopsystem poles will move in a -stable region, inspite of the changing system transfer functioncoefficents by the perturbations. First of all, themathematical model of a dc servo machine isobtained. Depending upon the varying systemtransfer function, design equations of thedynamic compensator are found out. Theseequations includes the ways of choosing thecontroller coefficents KP, KI, KD  which willgive a desired response for the tracking system.1. Introduction   The PID controllers are standard tool forindustrial automation. The flexibility of thecontroller makes it possible to use PID controlin many situations. The controllers can be usedin cascade control and other controllerconfigurations. Many simple control problemscan be handled very well by PID control,provided that the performance requirements arenot too high. The basic PID controlleralgorithm is,	 tdiP dtdeTdsseTteKtu0)(1)()(             (1)where u is the control variable, e is the errordefined as e=r - where r is the referencespeed value and  is the motor output speed.   To get  the best performance by thecontroller algorithm given by the equation(1), it is obvious that we must know thesystem parameters very well. In the realworld, the system parameters vary as thesystem changes its behaviour by theenvironmental changes, temperaturevariations, friction and load. To get thedesired response, these variations must betaken under consideration.[7]   In this paper, dynamic controller of a dcservomotor which is 20V and coupled to aload through a coupler is implemented by adigital signal processor. The most importantpart which makes it different from othercontrollers is the use of a fast processorwhich works parallel with a very fast ADC(analogue-digital converter). How it changesthe system response by fast conversion of aanalogue signal and computation of thecontrol algorithm  is examined in this work.It is chosen 8kHz for sampling frequencyfrom the tacho-generator. The controller isdesigned to make the steady-state errorminimum. Compensator coefficients arechosen to satisfy the robustness of thesystem by detaining the closed-loop systempoles in a closed stable -region. Polecancellation and changing the dynamics ofthe system by adding new poles and zeros tothe system is implemented by the processor.The processor used in this work is, 40MhzTMS320C50 DSP with 10k x 16 words ofon-chip RAM which works parallel withTLC320C40 analogue interface circuitry (AIC)with 14 bit resolution [3]. The processor iscommunicated with a PC through RS232 serialport. Power Electronics circuit is formed by adrive circuit, power mosfet and opto isolatorwhich separate the low voltage network fromthe drive circuit. The motor is controlled by amosfet which works in chopper mode.   The structure of the note is as follows.Section II gives the mathematical model of a dcservo motor. Section III describes therobustness of a feedback system and the -stability. Section IV gives the synthesisequations of the PID controller and discussesthe solutions of the synthesis problem of thecompansator depending on the limitations onthe -stable region. Section V gives the idea ofcomputations of the controller on a digitalsignal processor.2. Mathematical model of a dc servo      motor  A  servomotor is a DC motor designedspecifically to be used in a closed-loop controlsystem. The circuit diagram of a servomotor isgiven in figure 1. In this figure ea(t) is thearmature voltage. Rm is the armature resistanceand Lm is the armature inductance. The voltageem(t) is the voltage generated in the armaturecoil because of the motion of the coil in themotor’s magnetic field and is usually calledback –EMF [4].Then;dtdKtem	)(              (2)where K is the motor parameter,  is the fieldflux and  is the angle of the motor shaft. Weassume that the flux does not change during theoperation.  HencedtdKte mm	)(   .              (3)Figure 1 Circuit diagram of a servomotorThe Laplace transform of (3) givesEm(s)=Kms(s)  .         (4)For the armature circuit we can writeEa(s)=(Lms+Rm).Ia(s)+Em(s) .         (5)The mathematical equation for thedeveloped torque is(t)=K1Ia(t)=Ktia(t) .         (6)Finally, for the mechanical terminal pair wecan write,dtdtdtdJ	 )(22        (7)where, J is the inertia connected to themotor shaft and  includes the frictioncoefficents. Solving these set of equationsfor the motor shaft angle yields [4]sJssTs	2)()(   .         (8)as a result of these equations, one can obtainthe motor transfer function)()()()(mmma KKRsJRKsEswsGtt 		      (9)3.  - stability and robustness analysis   In a -region which is enclosed by a borderof  can be expressed by,{s | s=	()+jw()   [ - ,  +] }. - and+ borders can be - or +.  parameter iscalled as the generalized frequency (figure 2).Figure 2 : -Stable region and the -stability border   Our desire is to place our  closed loop  polesin a -region which is on the left and side ofthe segment as shown in figure 3.   Poles of the feedback control system will bedetained by the segment whose mathematicalexpression can be defined byn	 	            (10)and if we solve equation (3.1) for ,)1( 222  	 n   .            (11)Figure 3 :  Desired Closed loop polelocationIt is obvious that,21 	 	                   (12)and the border equation can be expressed as,		21		 jjs     (0) .       (13) Because of the limitations of  and21sin 		    ,                   (14) can vary up to equation (3.5). If we choosethe generalized frequency as    andexpress the border by the generalizedfrequency then we obtain21	 js       (-a) .       (15)jwsdG borderG Regionjwsq angledG stabilityborder-1Specially for the value of 21	  the borderequation will be,		 jjS 		)(            (16)3.1 Ackermann, Koesbaurr theoremIf we express the characteristic polynomial by  )(..1),( qassqs n rrv 	  then theroots of the polynomial for the border willprovide [5]Q• m()L	 qv{ | 0)),()(( 	 qjwr	 } .(17)For [-,+]  and )( ýmq rit must providethe following equation,-)(..)(0)(..)()(1010nnddddd)(qrr = 00                                  (18)where1)(0 	d)(2)(1 	 	d..)())()(()()(2 1221 		 -+ 	 iii dwdd(i = 1,2,..,n-1) .           (19)The proof of the equations expressed in (18)and (19), are well explained by Ackermann in[5].4. Sythesis Inequalities for the controller   The characteristic equation of the automaticcontrol system which is given by the blockdiagram in figure will be,023	TJRKKsTJRKKKKKKRsTJRJRTKKTKKTRsmImDpmmmmpmmtttttt                                                        (20)In generalized form (20) can be expressed as001223 	 asasas  and if the motorparameters vary in a given intervalas ),(+-  ,  Kr  constant ,  Rmconstant  ,  Kp = constant  ,  KI=constantand finally J(J- , J+). Then the coefficentsof the characteristic polynomial will movebetween a region defined by, a2- a2a2+and a1-a1a1+  respectively. If we choose KIsuch that it will satisfy,1	TJRKKmI t  then KI will be determined by,tKTJRK mI 	       (21)If we use the equations (18) and (19)respectively then we will get the followingrelations between a1, a2  and . 1212 231 		a                   (22)22122	a       (23)The plot of the given equations of (22) and(23) will be as shown in figure 4. Toimplement the nonlinear relations by asignal processor, a convex region must bechosen on the plot. We will try to push thecoefficents of the transfer function into theconvex region shown in figure 4.Figure 4: Desired -Stable region for theclosed loop transfer function coefficents To do this, we will use the compensatorcoefficents, then we will get the followinglimitations on choosing the compensatorcoefficients. If we use the worst coefficients inthe  given transfer function (20); we shouldchoose the compensator coefficients as shownby (4.5) and (4.6) respectively.TKRJTKKTRTRJaKTKRJTKKTRTaRjmmmmdpmmmdmtttt++-+---+22(24)ttttttKKKKKRTRJaKKKKKKRTRJapmmmdDpmmmd++-+--+-11   (25)5. Conclusion During the implementation process;  thecontroller coefficents are computed and thecompensator in Laplace domain is found out,then by Z-Transform the compensatorequivalent is computed in digital domain andprogrammed by DSP. As a result it has seenthat the controller forces the motor to moveas the reference speed and the controllerdoes not miss the domination on the motoras the motor coefficients vary by thedisturbances. Reference speed is obtained bysoftware and the output speed of the systemis measured by a digital tachometer.Figure 5: System response to thereference speedReferences[1] DSP Solutions for BLDC MotorsApplication Report . TexasInstruments, 1997[2] TMS320C5x User’s Guide. TexasInstruments, 1997[3] TMS320C5x DSK Application Guide.Texas Instruments, 1997[4] Charles L. Phillips, Royce D. Harbor“Feedback Control Systems” 1998Prentice –Hall Inc.[5]  Jürgen Ackermann “RobusteReguling”, Springer 1993[6] Katsuhiko Ogata “Discrete-TimeControl Systems”, Prentice Hall1987[7] Karl J. Astrom, Björn Wittenmark“Adaptive Control”,1995 Addison-Wesley Publishingad2+ad2-a2a1a=-¥a=0dG Borderad1- ad1+Desired regionfor the varyingsystemcoefficients

Synthesis inequalities for the robust PID controller for a DC Servo motor and Implementation by using digital signal processor

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Synthesis inequalities for the robust PID controller for a DC Servo motor and Implementation by using digital signal processor

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