ET 438a Control Systems Technology Laboratory 4 Modeling Control Systems with MATLAB/Simulink Position Control with Disturbances

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1 ET 438 Control Systes Technology bortory 4 Modeling Control Systes with MATAB/Siulink Position Control with Disturbnces bortory erning Objectives After copleting this lbortory you will be ble to:.) Convert differentil equtions representing n electroechnicl control syste into block digr with feedbck..) Use MATAB Siulink softwre to represent control syste. 3.) Ipleent proportionl nd proportionl-integrl controller using Siulink blocks. 4.) Observe the effects of disturbnces on feedbck controller response. 5.) Copre the responses of proportionl nd proportionl-integrl controllers with respect to response tie nd stedy-stte error. Theoreticl nd Technicl Bckground A well designed utotic control systes reduces the effects of outside syste disturbnces for fixed set-point. A coonly encountered exple of control syste tht responds to outside disturbnces is the cruise control found in utoobiles. The driver estblishes cruising speed nd the controller responds to chnges in rod conditions to intin this desired vlue. One wy to represent n outside disturbnce is to consider it s n input to the syste tht dds to the syste process lod. This lb ctivity odels n electroechnicl ntenn positioning syste shown in Figure. This figure shows pernent gnet dc otor driven by n djustble dc power supply djusting the ngulr position of n ntenn rry. The desired position of the ntenn s function of tie is given by A (t). The wind velocity produces torque on the positioning shft tht is reflected to the otor shft through speed reducing gerbox. This gerbox reduces otor speed nd increses otor torque llowing the otor to effectively use its echnicl power output to position the ntenn rry. A vrible high-power dc power supply djusts the otors rture voltge tht, in turn, chnges the ntenns position through the gerbox. An utotic ntenn controller tkes reference ntenn position, sp (t) s n input. This is usully constnt with respect to tie but y be liner function for trcking ppliction. Fll 06 lb4_et438.docx

2 Wind velocity A (t) T W (t) T (t) w (t) Ger reduction Dc Motor v (t) Motor Dc Supply Antenn Control Antenn Desired position sp (t) Figure. Antenn Positioning Syste with Externl Wind Disturbnce. This lb ctivity uses MATAB Siulink to copute ntenn ngle, A (t) for given disturbnces cused by wind velocity chnges using block digr odel. The lb ctivity will ipleent proportionl only control nd proportionl-integrl control. Copring the resulting ntenn output position using these two types of controllers will deonstrte the dvntges nd disdvntges of ech control ode. Mtheticl Modeling of the Positioning Syste-Dc Motor This control syste ctutor is the rture-controlled pernent gnet dc otor. Figure shows the dynic odel of the rture controlled otor. This odel ssues tht the otor drives echnicl lod directly fro its shft. i (t) B J w (t) T (t) B J R e b (t) T (t) v (t) Figure. Arture Controlled Dc Motor Model with Direct od Drive. Fll 06 lb4_et438.docx

3 The odel preter definitions re: v (t) = rture voltge (V) i (t) = rture current (A) e b (t) = otor bck EMF (V) T (t) = otor developed torque (N-) w (t) = otor shft speed (rd/sec) T (t) = echnicl lod torque (N-) R = rture resistnce (Ohs) = rture inductnce (H) B = otor viscous friction (N--sec/rd) B = lod viscous friction (N--sec/rd) J = otor rottionl inerti (kg- ) J = lod rottionl inerti (kg- ) This representtion requires two ore preters to coplete the odel. These preters re: K e = bck EMF constnt (V-sec/rd) K T = torque constnt (N-/A) When using SI units (rd/sec nd N-) the nuericl vlues of K e nd K T re equl. The follow syste of equtions represents the dynic response of the otor speed to chnges in rture voltge. In these equtions, the otor rture current nd the speed copletely define the otor lod response. The rture voltge nd the lod di (t) R dt dw(t) dt KT J J K e i (t) w i (t) v (t) (t) () B B T (t) (t) J J w J J (b) () torque re the inputs to the syste. Solving these differentil equtions, clled stte equtions in control syste theory, gives plots of otor rture current nd speed with respect to tie. Eqution () represents the electricl dynics of the dc otor nd includes n inductive tie constnt tht restricts the rte t which current cn chnge in the otor rture. Siulink uses equtions in this for odified for nuericl integrtion in coputer to produce response plots. Eqution (b) describes the echnicl dynics. This eqution includes echnicl tie constnt represented by the rtio of totl viscous dping (B B ) to Fll 06 3 lb4_et438.docx

4 totl rottionl inerti (J J ). This rtio deterines how quickly the otor/lod speed cn chnge. Equtions () cn be rerrnged nd expressions () nd (3) included giving the following syste of equtions. v (t) T (t) K i (t) () T e (t) K w (t) (3) b e di (t) dt eb(t) i(t) R (4) dw (t) T (t) J J B B w (t) T (t) (5) dt Tking the plce trnsfor of these equtions gives the syste of equtions below. These equtions for input/output reltionships used to produce the block digr shown in Figure 3. T J B b eq eq (s) K (s) J I (s) (s) s B V (s) E J J Direct coupled only B B Direct coupled only T e eq E (s) K w I (s) s R eq T b (s) (s) T (s) Figure 3 shows tht the rture-controlled dc otor hs n iplicit internl negtive feedbck. There is n increse in rture current whenever the otor rture speed decreses. This cuses the torque to increse nd the rture speed to increse to new vlue. This feedbck ccounts for the lost constnt speed chrcteristic of the seprtely excited nd pernent gnet dc otors. Figure 3 shows tht the difference between rture voltge nd bck EMF produces voltge difference tht deterines the rture current. The vlues of nd R deterine the gnitude nd rte of chnge of this current. The torque constnt converts rture current into developed otor torque. Torque produces the otor shft speed through the echnicl dynics. The totl viscous friction liits the otor Fll 06 4 lb4_et438.docx

5 T (s) V (s) - /( sr ) K T - /(J eq sb eq ) (s) I (s) T (s) K e Figure 3. Block Digr of n Arture-Controlled dc Motor with Direct Coupled od. speed while the totl rottionl inerti deterine the rte of speed chnge. This controller onitors the position of the ntenn, so position ust be relted to the otor speed. Eqution (6) uses the rte of chnge in position to express otor speed. d dt w (t) (6) Where = otor shft position (rd) This eqution shows tht integrting the speed of the otor shft will give the otor shft position. Figure 3 cn then be odified to deterine otor shft position by dding integrtion to its output. Integrtion in the plce doin is division by the plce vrible s. Figure 4 shows the otor odel for rture voltge input nd shft position output. T (s) V (s) - /( sr ) K T /(J eq sb eq ) (s) /s (s) I (s) T (s) K e Figure 4. Arture-Controlled Dc Motor with Externl od nd Shft Position Output. Fll 06 5 lb4_et438.docx

6 Eqution (7) now shows the stte eqution representtion for the dc otor with position output for directly coupled echnicl lod. di (t) R dt K e i (t) w dw (t) K T i (t) dt J J d(t) w(t) (c) dt v (t) (t) () B B T (t) (t) J J w J J (b) (7) In ost cses, the electricl tie constnt given by e = /R is uch sller thn the echnicl tie constnt =J /B. This siplifies the trnsfer function nd reduces the coplexity of the theticl odel. If the rtio of the tie constnts, / e 00 use the reduced order odel given by eqution (8) below. (s) K V (s) s s s Where K s s R R KT B K R J B K E E K K T T (8) The reduced odel eliintes the torque sution node shown in Figure 4. A speed disturbnce ust now represent the wind torque disturbnce. Figure 5 shows the block digr for reduced order otor odel tht hs n externl disturbnce input. T (s) /(J sb ) V (s) K s /( s s) (s) /s (s) (s) (s) Figure 5. Reduce Order Dc Motor Model with Externl lod. The echnicl dynics given by the dping nd rottionl inerti of the externl syste trnsfor torque disturbnce into corresponding speed disturbnce. Fll 06 6 lb4_et438.docx

7 This version of the otor lod odel seprtes echnicl lod preters fro the otor electricl preters. The lod rottionl inerti nd viscous friction re the preters tht convert the lod torque into the lod speed. The resulting otor-lod speed produces position chnge fter integrtion. The order reduction lso reoves the internl feedbck loop between the internlly genertor voltge nd the terinl voltge tht controls the rture current. The block digr shows the otor rture voltge controlling the otor speed directly through the first-order block. Mtheticl Model of Ger Systes The ntenn syste otor drive couples to the ntenn through speed reducing ger syste. The ger rtio chnges the speed nd torque vlues produced t the otor shft. This rtio lso chnges the gnitude of the viscous friction nd rottionl inerti tht the otor shft sees. Figure 6 shows representtion of siple ger drive coprised of two gers. The echnicl power source delivers speed T w N T w N Figure 6. Speed, Torque nd Position Representtion For Gers. w nd torque T. The output is speed w nd torque T. The drive oves the input ger over rdins while the output ger oves over distnce of rdins. The rdius of ech ger deterines the ngulr distnces ech ger covers per unit tie. This lso deterines the ngulr speed. The gers cnnot slip so the distnce covered by the driven ger ust equl the distnce covered by the output ger. The following eqution gives the length of rc covered per unit tie in ters of the ger rdius. d dt d dt r r (9) Fll 06 7 lb4_et438.docx

8 Where: r = rdius of driven ger r = rdius of output ger The derivtive of the ngulr displceent is the ngulr speed in rdins/second. Also, the nuber of teeth in the ger is proportion to the rdius. Mking these substitutions gives the following expression for speed chnge. w N w w N N w N (0) Where N = nuber of teeth in the driven ger N = nuber of teeth in the output ger w = drive speed w = output speed The position of the output ger shft follows siilr reltionship. N N () Where = drive shft position = output shft position The torque rtio is the inverse of the speed rtio. As the output speed decreses, the output torque increses for fixed echnicl power input. This intins echnicl power blnce cross the ger box ssuing no friction or other losses. T N () T N Where: T = output torque T = drive torque Ger boxes chnge the vlues of viscous friction nd rottionl inerti seen by lods nd echnicl power sources. Figure 7 illustrtes typicl echnicl syste with viscous friction vlues B nd B nd rottionl inertis J nd J connected through ger box with N drive teeth nd N output teeth. The ger rtio chnges the vlues of echnicl output viscous friction nd rottionl inerti through the following foruls. Fll 06 8 lb4_et438.docx

9 J B eqd eqd J B N N N N J B (3) The preters J eqd nd B eqd re the totl equivlent rottionl inerti nd viscous friction seen fro the drive side of the ger box. w T J B N Drive B J N w Output Figure 7. Ger Box, Mechnicl od Inertis nd Viscous Friction. The ger rtio cn chnge torque, speed, nd position of the output shft bsed on equtions (0), () nd (). The following equtions specify output speed torque nd position in ters the drive vribles. N w N N N N T N w T () (b) (c) (4) When the output vlues re known, solve the bove equtions for the vlues of w, nd T to get the equivlent speed, position nd torque on the drive side of the ger box. An idel ger box is nlogous to the idel electricl trnsforer. Mechnicl power is equl on both sides of the ger box. The ger rtio reduces speed but increses torque to intin the power blnce on both sides of the device. The preters of viscous friction nd inerti reflect through the ger box by ultiply by ger rtio squred when Fll 06 9 lb4_et438.docx

10 viewed fro the drive side of the device. These preters re divided by the ger rtio squred when ll quntities re referred to the output side of the ger box. Position Sensing nd Negtive Feedbck T (s) sp (s) - e (s) Controller Motor nd Diver Power Supply A (s) (s) Position Sensor Figure 8. Position Sensing nd Error Genertion. Figure 8 shows the negtive feedbck loop required to generte n ntenn position error signl. The position sensor onitors the ntenn position nd y plify its output to scle it correctly for coprison with the set point position signl. The output of this block feeds to suing junction tht copres the esured position vlue to the desired vlue. An lgebric eqution shown below describes this process. K s (s) (s) (s) (5) sp e Where K s = sensor gin (could be.0 to siplify odel) sp (s) = setpoint vlue of ntenn ngle (s) = esured vlue of ntenn ngle e (s) = ngle error When K s =.0 the output of the sensor is (s) = A (s), the esured vlue equls the ctul ntenn position. The error ngle provides the input signl to the controller tht will then produce the correction signl for the dc otor/drive. Control Mode Models Three control odes exist for odifying the error signl of negtive feedbck controller. The odes re proportionl (P), integrl (I) nd derivtive (D). Cobining these odes give severl cobintions of control response with differing chrcteristics. Coonly found cobined odes include PI, PD nd PID controls. Fll 06 0 lb4_et438.docx

11 Adjusting the constnts ssocited with these cobintions "tunes" the controller nd odifies the syste's response. The proportionl control ode tkes the input error signl nd plifies it by constnt. Using this control ode y result in lrge plifiction fctors to chieve n cceptble stedy-stte error tht cn result in n unstble syste. The integrl control ode sus the error signl over tie nd cn drive the stedy-stte error of syste to zero over tie. The derivtive control ode cnnot be used lone since it only produces nd output when the error input is chnging. The derivtive ode produces correction signl bsed on the error's rte of chnge. This produces n nticiptory ction tht cn iprove syste s stbility. This lb ctivity will focus on proportionl (P) control nd proportionl-integrl (PI) control odes. Figure 9 shows how to odel the P nd PI controller in Siulink. In these representtions, K p = proportionl gin, K I = integrl gin nd s is the plce vrible. E(s) V(s) K p Proportionl Controller Model E(s) K p K I /s V(s) E(s) K V(s) I K s p K P s Alternte PI Controller Model Proportionl-Integrl Controller Model Figure 9. Siulink Block Digr Models for P nd PI Controllers. The figure shows two lterntives for relizing the PI controller s block digr odel. The first PI odel uses suing junction to cobine the P nd I ctions while the second representtion shows n lgebriclly reduced version using two cscded blocks. The PI controller introduces single pole in the syste locted s s=0 nd zero t s=-k I /K p. Adjusting the vlues of K p nd K I odifies the syste response nd eliintes the stedy-stte error. In systes tht lredy include n integrtor, the ddition of pole t zero reduces the trcking error to zero. The ntenn controller syste includes n integrl ction in the conversion of otor shft speed into shft position. Using PI control should produce trcking error of zero fter soe tie dely. Negtive Feedbck nd Disturbnce Rejection One of the benefits of utilizing negtive feedbck is tht the controlled syste tends to return to its desired vlue fter encountering n outside disturbnce on the output signl. Figure 0 shows the block digr of syste with disturbnce injected on the controller output. An externl disturbnce, D(s) cts on the syste output, Y(s) to Fll 06 lb4_et438.docx

12 ove the syste fro stble vlue deterined by the syste setpoint, R(s). The overll response cobines the influence of both inputs on the syste. D(s) R(s) E(s) - Controller G (s) G (s) Y(s) H(s) Figure 0. Block Digr of Negtive Feedbck With Externl Disturbnce. Superposition finds the overll output of the syste tking R(s) nd D(s) s individul inputs with the other input set to zero. Evluting the resulting block digrs for ech individul input signl with the other input set to zero gives the output response produced by ech input. The su of the two outputs gives the totl syste response to both the disturbnce nd the setpoint. Eqution (6) shows how the set point nd the disturbnce signls contribute to the totl output for proportionl controller with gin of K p. Kp G (s) G (s) G (s) Y(s) R(s) D(s) (6) Kp G (s) G(s) H(s) Kp G (s) G(s) H(s) This eqution shows tht if the vlue of K p >>, then the rtio ultiplying R(s) pproches one. This ens tht the output will trck the input with no stedy-stte error. The second ter shows tht for proportionl gin, the gnitude of the disturbnce diinishes s the vlue of K p becoes very lrge. Eqution (7) shows the se reltionship s (6) only with proportionl-integrl (PI) controller used. The PI controller dds pole t s=0 nd zero t s=-k I /K p. K Y(s) p (s ) G(s) G (s) sg(s) R(s) D(s) s GH(s) s GH(s) Where GH(s) K K I / K p P (s ) G (s) G (s) H(s) (7) Fll 06 lb4_et438.docx

13 Eqution (7) hs two preters, K p nd K I, tht control syste response. Setting the zero of the PI control, which is the preter =K I /K p, to vlue less thn the doinnt pole of the open-loop syste chnges the syste perfornce. The doinnt pole is the trnsfer function denointor polynoil root with the sllest nuericl vlue, which is the longest tie constnt. This cncels soe of the doinnt pole s effects nd cn increse the syste response speed. Setting the vlue of controller zero t exctly the vlue of the dointe pole, in this cse cuses, the syste to oscillte for ll vlues of K p. Setting the zero vlue to be greter thn the dointe pole cuses the output to be unstble. This nlysis ssues the disturbnce is zero. The roots of the chrcteristic eqution ust ll hve positive negtive rel prts for syste stbility. The chrcteristic eqution for the PI controller is given by s p K (s ) G (s) G (s) H(s) 0 (8) Mintining constnt while djusting K p requires chnge in K I. For fixed zero loction, then this reltionship holds. K (9) I K p For the PI controller selecting vlue of nd K p will djust the syste response. Once stble vlue of is found the vlue of K p is djusted to iprove control perfornce. Fll 06 3 lb4_et438.docx

14 T w (s) N /N Desired Angle sp (s) Angle Detector Motor Gin Dc Supply V (s) 00 K p K T - - Q E (s) Angle error V c (s) P Controller R s I A (s) T M (s) Jeq B eq s Jeq M (s) K e A (s) N /N /s Antenn Angle A (s) Position Sensor Antenn velocity.0 Figure. Block Digr of Antenn Positioning Syste Using Full Arture Controlled Dc Motor Model nd Proportionl Controller. Fll 06 4 lb4_et438.docx

15 Desired Angle sp (s) T w (s) PI controller V c(s) s K p E(s) s K N /N I K p Angle Detector Gin E(s) V c (s) V (s) 00 V c (s)/e(s) K T - - R s I A (s) Q T PI Controller Motor M E (s) (s) Angle error Dc Supply Jeq B eq s Jeq M (s) K e A (s) N /N /s Antenn Angle A (s) Position Sensor Antenn velocity.0 Figure. Block Digr of Antenn Positioning Syste Using Full Motor Dc Model nd Proportionl-Integrl Controller Fll 06 5 lb4_et438.docx

16 T (s) J B s J (s) N /N Desired Angle sp (s) - Angle Detector Gin E (s) Angle error 00 K p V c (s) P Controller Motor Dc Supply V (s) K s /( s s) (s) (s) (s) (s) N /N A (s) /s Antenn Angle A (s) Position Sensor.0 Figure 3. Antenn Positioning Syste Using Reduced Order DC Motor Model nd Proportionl Controller. Fll 06 6 lb4_et438.docx

17 PI controller V (s) s c K p E(s) K I K p s T (s) J B s J (s) N /N Angle Detector Desired Angle Gin sp (s) E(s) 00 - V c (s)/e(s) V c (s) V (s) K s /( s s) (s) (s) N /N /s Antenn Angle A (s) Q E (s) Angle error PI Controller Motor Dc Supply Antenn & Disturbnce velocity Position Sensor A (s) (s) (s).0 Figure 4. Antenn Positioning Syste Using Reduced Order DC Motor Model nd Proportionl-Integrl Controller. Fll 06 7 lb4_et438.docx

18 b 4 Procedure The ntenn positioning syste shown in Figure hs the following preters surized in Tble. Use these preters to copute the vlues of the constnts in the block digrs shown in Figures through 4. Tble - Syste Preters Preter Preter Description (Units) Preter Vlue Motor Arture Inductnce (H) 0.00 R Motor Arture Resistnce (Ohs) 4.0 K T Motor Torque Constnt (N-/A) 0.4 K e Motor EMF Constnt (V-s/rd) 0.4 B Motor Viscous Friction (N--s/rd) 0.00 J Motor Rottionl Inerti (kg- ) 0.00 K pw DC supply voltge conversion constnt N Motor Ger Tooth Count 5 N Output Ger Tooth Count 650 B Antenn Syste Viscous Friction (N--s/rd) 37.5 J Antenn Syste Rottionl Inerti (kg-s ) 50 Prt : Proportionl Controller with Detiled Motor Model. Use the preters fro Tble bove to copute the constnts in the block digr shown in Figure. Enter the results of your clcultions into Tble A- in the ppendix for lter subission.. Open Siulink odel using MtAB nd construct the block digr odel of Figure using this progr. The setpoint input hs the following grphicl representtion nd theticl description. Refer to the lb video presenttions to see how to sp 0.5 t (t) t t 5 Angle Setpoint t sec Tie (seconds) Fll 06 8 lb4_et438.docx

19 construct this wvefor using Siulink inputs. Use this input for ll the following block digrs in this lb. 3. Construct the torque disturbnce shown in the block digr using the following theticl description nd figure. This input is squre pulse of two seconds beginning t 5 seconds nd ending t 7 seconds. It hs gnitude of 0 N-. (t) T W t 5 5 t 7 7 t 5 Torque 0 5 sec Tie (seconds) Refer to the lb video presenttions to see how to construct this wve shpe using Siulink inputs. Use this disturbnce for ll following lb siultions. 3. Connect the setpoint input nd the disturbnce inputs to the block digr constructed using Siulink nd set the vlue of K p to 0.5 nd the siultion stop tie to 5 seconds. 4. Review the lb video presenttion on ultiplexed scope disply. Set up scope nd ultiplexer to gther both the setpoint wve nd the output wve siultneously. Ne the connections entering the ultiplexer to tch the signl source. 5. Set the scope to sve its cquired dt to the MtAB workspce. View the video presenttion tht describes this process. 6. Run the ntenn siultion for K p =0.5 nd review the results by double clicking on the scope icon in Siulink. 7. Downlod nd sve the MtAB script file ned lb4process. to your coputer or storge device. Add the pth to this file s loction to the MtAB environent. 8. Enter the MtAB cond window nd type the script ne exctly. MtAB is cse sensitive. 9. A propt will pper sking for the vlue of K p used for this siultion. Enter the correct vlue nd hit enter. 0. The script produces two plots: A tie plot of the syste input nd output on the se xis nd n error plot with the xiu error listed nd the tie tht it occurs.. Copy these plots to Word docuent nd sve the for lter review nd subission. 7 sec Fll 06 9 lb4_et438.docx

20 . Run the ntenn siultion using K p = nd review the results by double clicking on the scope icon. 3. Enter the MtAB cond window nd type lb4process on the cond line to execute the script. Enter the vlue of K p when propted. 4. View the tie plots produced nd copy the to Word docuent for lter review nd subission. 5. Repet steps to 4 for the following vlues of K p :, 4, 8, nd 6. Note how the vlue of K p chnges the xiu error. Prt b: Proportionl-Integrl Controller with Detiled Motor Model. Use the preter vlues of Tble to copute the constnts for the block digr odel shown in Figure replcing the proportionl controller with one of the PI controller structures shown in Figure 9. Use the se inputs s the previous odel.. Set up scope to cpture the setpoint input nd the controller output just s in Prt of this lb. Set the scope to sve its dt to the MtAB workspce for ore processing. 3. Set the vlue of K p to 4 nd the vlue of = Set the siultion tie to 5 seconds nd run the siultion. 4. Enter the MtAB cond window nd type lb4process on the cond line to execute the script. Enter the vlue of K p when propted, which should be 4 in this cse. 5. View the tie plots nd copy the to Word docuent for lter review nd subission. 6. Repet steps 3 to 5 for =0.4 nd 0.. Mintin the vlue of K p t 4 for these two siultions. 7. Repet steps 3 to 5 for =0. nd K p =8. Does this chnge the vlue of the xiu error? 8. Note the differences in the error between the proportionl only nd proportionlintegrl controllers perfornce. Prt : Proportionl Controller with Reduced Motor Model. Use the preter vlues in Tble nd the foruls for the reduced dc otor odel to copute the constnts Figure 3 s block digr. Plce the coputed vlues into Tble A- in the ppendix for future subission.. Copute the rtio of the otor electricl to echnicl tie constnts =J /B nd e = /R to the criteri listed in the lb theoreticl nd technicl section: / e 00. Does this otor qulify for odel reduction? Fll 06 0 lb4_et438.docx

21 3. Use Siulink nd construct this block digr using the constnts fro step. Note tht the ntenn echnicl dynics convert the disturbing torque into speed through the inverse of the ger rtio. Use the se inputs s those in Prts nd b. 4. Set up scope to cpture the syste input nd output plots. View the instructionl video tht shows how to coplete this tsk. Set the scope to sve the siultion s dt points to the MtAB workspce for lter processing. 5. Set the siultion tie for 5 seconds. Run siultions of the proportionl controller using the reduced odel nd K p =0.5. View the results on the scope when the siultion ends. 6. Enter the MtAB cond window nd type lb4process on the cond line to execute the script. Enter the vlue of K p when propted, which should be 0.5 in this cse. 7. View the tie plots nd copy the to Word docuent for lter review nd subission. 8. Repet steps 5-7 for K p =,, 4, 8, nd 6. Note how chnging the vlue of K p ffects the gnitude nd tie of the xiu error. Copre this to the full dc otor odel response. Prt b: Proportionl-Integrl Controller with Reduced Motor Model. Chnge the Siulink odel to include the Proportionl-Integrl controller shown Figure 4 s block digr. All other constnts should be the se s used with the proportionl only controller of Prt. A scope should rein connected to the input nd output points s in Prt. This scope should be set to sve dt to the MtAB workspce for ore processing.. Set =5.9, K p =4, nd siultion tie to 5 seconds. Run the siultion nd exine the scope output showing both the desired input nd the position output. Does this syste output pper stble or does it oscillte? 3. Enter the MtAB cond window nd type lb4process on the cond line to execute the script. Enter the vlue of K p when propted, which should be 4 in this cse. 4. View the tie plots nd copy the to Word docuent for lter review nd subission. Copre this to the proportionl only controller output. Does the error stbilize for this configurtion? 5. Repet steps -4 for K p =4 nd =3 nd. Note how chnging the vlue of the preter chnges the syste response. Copre these plots to those generted using the full dc otor odel. Fll 06 lb4_et438.docx

22 b 4 Assessent Subit the following ites nd perfor the listed ctions to coplete this lbortory ssignent.. Copile the results of the Prt siultions in Tble A-3 in the Appendix. Use the plots generted for error nd syste response to coplete this tble for ech vlue of K p listed.. Copile the results of Prt b siultions in Tble A-4 in the Appendix. Use the plots generted for error nd syste response to coplete this tble. 3. Copile the results of the Prt siultions in Tble A-5 in the Appendix. Use the plots generted for error nd syste response to coplete this tble for ech vlue of K p listed. 4. Copile the results of Prt b siultions in Tble A-6. Use the siultion results to coplete this tble. Indicte if the response is stble or unstble by plcing S or U in the lst colun of the tble. 5. Convert Appendix A into pdf docuent nd subit it to the b 4 ssignent Dropbox in the course online terils. 6. Coplete the online quiz ssocited with b 4 in the course online terils. Use ll dt nd plots gthered nd the lb hndout s reference during the quiz. Fll 06 lb4_et438.docx

23 Appendix A Clcultions Sury nd Siultion Results Tble A-: Prt ndb Model Constnts Forul/preter Nuericl Vlue B eq J eq / /J eq R / B eq /J eq N /N Tble A-: Prt nd b Model Constnts Forul/preter Nuericl Vlue B J K s /J s B /J N /N N /N Tble A-3: Prt Proportionl Control Full Dc Motor Model Error Results K p Mxiu Error Tie of Mxiu Error Tble A-4: Prt b Proportionl-Integrl Control Full Dc Motor Model Error Results K p Mxiu Error Tie of Mxiu Error Fll 06 3 lb4_et438.docx

24 Tble A-5: Prt Proportionl Control Reduced Dc Motor Model Error Results K p Mxiu Error Tie of Mxiu Error Tble A-6: Prt b Proportionl-Integrl Control Reduced Dc Motor Model Error Results K p Mxiu Error Tie of Mxiu Error Stble/Unstble (S/U) Fll 06 4 lb4_et438.docx