TU/e. Eindhoven University of Technology Department of Mechanical Engineering Control Systems Technology Group

Transcription

1 TU/e Eindhoven Univerity of Technology Department of Mechanical Engineering Control Sytem Technology Group Modelling and control of ledge and optical pickup ytem W.H.T.M. Aangenent DCT.XX June Supervior: Prof.. J. Stoutrup P. Anderen E. Vidal Sanchez prof. dr. ir. M. Steinbuch

2 Abtract In thi report, a model ha been developed for a part of a CDM- Indutrial module ued in the B&O BeoSound ytem, namely the ledge and radial ytem. The influence of movement of the ledge on the radial error ignal ha been invetigated and a imulation ha been done with the nowaday ued control trategy. Thi trategy i baed on puling the DC-motor that i driving the ledge until it tart to move. Thi lead to a very nonmooth movement of the ledge and alo there i a poibility that the radial actuator will aturate which lead to big error. To generate a more mooth movement of the ledge, a controller ha been deigned, coniting of a PD controller and a feed forward part, that make the poition of the ledge follow a reference. Unfortunately, the poition of the ledge i not meaured and therefore thi ignal cannot be ued a feedback. The ledge ytem i ued in the firt place to keep the radial control ignal from aturating o a natural choice for a feedback ignal i thi radial control ignal. Another PD controller with feed forward i deigned which track a reference radial control ignal when thi i needed. Since there i backlah and tiction preent in thi ytem, there are ome retriction on the bandwidth of thi controller. Becaue of the tiction, the bigget diturbance on the radial ytem i generated when the ledge tart to move. To decreae thi error, another radial controller can be ued which ha a higher bandwidth. Thi gain cheduling approach ha alo been added to the total ytem and imulation are done.

3 Content. Introduction 4. Background.. 4. Goal The ytem 4.4 Problem decription..4.5 Outline of the report. 4. Sytem decription 5. Structure of the compact dic.. 5. Optical ytem. 6.3 Focu and radial enor The ervo mechanim in the CD-player Modelling of the radial and the ledge ytem 3. Model of the radial actuator. 3.. Electrical part of the radial actuator. 3.. Mechanical part of the radial actuator. 3. Model of ledge ytem DC-motor Electrical part Mechanical part Gear tranmiion The ledge 6 4. Deign of radial controller and pule teered ledge controller 9 4. Controller for the radial loop Pule teered controller for the ledge ytem 5. Deign of a continuou controller 5. Control trategy for the ledge. 5. Sytem and controller Uing the radial control ignal Reference ignal Sytem and controller Reult Gain cheduling 7 6. Concluion and recommendation 9 Reference 3 Appendice: A. Value radial actuator and ledge ytem. 3 A. Computation of the gear ratio... 3 B Simulink model of the ledge ytem.. 33 C Controller 38 3

4 . Introduction. Background In the deign of optical torage, playability and low manufacturing cot play an important role. Unfortunately, thee two are each other oppoite. In order to guarantee good playability, good controller and precie mechanim hould be implemented. In the Beoound 9 ytem deigned by Bang & Olufon, a ledge ytem i preent which make ure that the radial controller operate in it linear range. Thi ledge hould move a the radial actuator almot aturate. Nowaday, the ledge i teered by giving electric pule, however, the diturbance caued by thi movement on the radial loop i not known. Becaue of the fact that in the ledge ytem friction, tiction and backlah are preent and meaurement of the poition of thi ledge are not available due to production cot, a continuou controller ha not uccefully been deigned.. Goal The goal of thi report i to make a model of the ledge ytem and the radial controller and to determine the influence of the movement of the ledge on the radial loop. Thi hould firt be done for the nowaday ued control trategy baed on pule and after that an effort hould be made to deign a controller which move the ledge more continuou and a few time a poible. Thi can be done by letting the ledge follow a mooth reference which bring the radial controller from one end of it range to the other..3 The ytem The ytem ued i a CD-player manufactured by Bang & Olufen, the BeoSound ytem and pecific the CDM Indutrial module. In chapter two the general CD-player ytem i decribed..4 Problem decription In order to generate a model of the total ytem, the variou ubytem have to be identified and modeled. A model ha to be made for the DC-motor, gear mechanim, ledge ma with friction and the radial actuator (len connected with a coil). A S- function ha to be written which imulate the pule teered control ignal and a mooth continuou controller ha to be deigned. Poition meaurement of the ledge are not available o another ignal ha to be ued for feedback..5 Outline of the report At firt, omething will be aid of the CD-player mechanim itelf. Chapter three cover the modeling of the radial actuator and ledge ytem including the DC-motor and the gear mechanim. In the next chapter a radial controller i deigned and the nowaday ued pule teered ledge controller i implemented. After thi, in chapter five, a continuou controller i deigned together with a gain cheduled radial controller. Finally, concluion and recommendation are preented in the ixth chapter. 4

5 . Sytem decription Thi ection contain a decription of the CD and the optical ytem together with a decription of the actuator and enor which contitute the CD-player ytem.. Structure of the compact dic A CD i a medium on which data i tored in digital form. Analogue muic i ampled and recorded and encoded on the CD uing Pule Code Modulation (PCM). For ucceful torage ome additional data i added uch a error correction, ynchronization and modulation. The torage area of the dic extend from the inner to the outer perimeter and the dimenion can be een in figure.. figure.: Structure of the CD The inner and outer part of the dic are ued for torage of the lead-in and lead-out data. The tored data i repreented on a piral formed track, which conit of pit and flat. The laer ee thee pit a bump on the dic urface while a flat level with the ret of the urface. The pit have different length ( µm) and they repreent the PCM data which can be read by a laer. The dimenion of the data track layout can be een in figure.. figure. : Dimenion of pit and track 5

6 When the data i read from the dic the laer pae through a tranparent layer that i approximately. mm thick. The effect of thi layer i that the laer beam i narrowed down from the dic urface to the ignal urface from.8 mm tot.7 µm (ee figure.3). Figure.3 : Laer beam focuing The advantage of thi effect i that a cratch or other dic irregularitie on the urface of the dic are only a fraction of their original ize at the ignal layer. The laer ued to read the data from the dic ha a wavelength of 78 nm in air and 5 nm in the CD cauing the angle of refraction between the air and the dic. The height of a pit i one fourth of the wavelength, o the light hitting a pit travel one half of a wavelength horter than the light that hit a flat (ee figure.4). Thi create a phae difference between the light reflected from the pit and the flat which reult in detructive interference. A pit reflect about 5 % of the original intenity and a flat about 9 %. Thi change in light intenity make the CD-player able to ditinguih between pit and flat and in thi way to read the data. figure.4: Reflection of the laer on a pit and a flat. The optical ytem The optical pickup ytem in the Philip CDM-I drive mechanim ue the 3-beam ytem. It conit of a main beam, which i ued for focuing and reading the data and two atellite beam which are ued for tracking of the radial poition. Figure.5 give a chematic overview of thi optical pickup ytem: 6

7 figure.5: Schematic overview of optical pickup ytem At firt the laer beam i generated by the laer diode. Then the following happen: The beam pae through a light diffraction that generate the two atellite beam. The beam hit a polarization prim where only the vertical light pae. The beam are converted to parallel beam by the colliion len. The beam are turned 45 by a one fourth wavelength. The beam are reflected by the dic and end back through the wave heet, which reult in another turning of 45 o that the beam are converted to horizontal outgoing beam. The beam hit the polarized prim again and are reflected to the photodiode on the pickup. The main beam i reflected to diode D D, D D and D D3. The two atellite beam are reflected on diode D S and D S (ee figure.6) Figure.6 how the arrangement of the photodiode ued in the optical pickup ytem. figure.6: Photo diode in optical pickup With thi etup the diode can be ued to generate the focu and radial error a explained in the next ection. 7

8 .3 Focu and radial enor In the 3-beam optical pickup ytem the mainbeam detector diode i ued for the generation of the focu error ignal and the atellite diode (ide beam detector) are ued to generate the radial error ignal. A the ditance between the objective len and the dic reflective urface varie, the focal point of the optical ytem alo change and the image projected by the cylindrical len change it harpne. Thi image projected on the mainbeam diode generate three current were the diode D D and D D are compared to determine whether or not the optical pickup i in focu while D D3 i ued to read the data on the piral track. Three cae can be ditinguihed which are howed in figure.7. Optimal focuing Focu point to cloe Focu point too far figure.7: Three cae of focuing In the firt figure the focuing i optimal and the current generated by D D and D D are equal and the focu error ignal e f = DD DD i zero. If the focu point i too cloe to the dic, more light fall on D D than on D D o the focu error ignal i poitive and when the focu point i too far away, thi error ignal i negative. Thi information can be ued a input to a controller that keep the optical pickup alway in focu. To detect if whether or not the optical pick i on track, the two atellite diode D S and D S are ued. For thi ytem alo three cae can be ditinguihed which are howed in figure.8. Dic figure.8: Three cae of tracking 8

9 In cae (a) the mainbeam i located too far to the left while in cae (b) the mainbeam i located too far to the right. A the three pot drift to either ide of the pit track, the amount of light reflected by the three beam that encounter more pit area i reduced while the light intenity reflected by the beam that encounter le pit area, i increaed. The relative output from the two atellite diode provide a tracking error ignal, er = DS DS. When the ituation i a depicted in figure (c), the two atellite diode receive the ame amount of light and the radial error ignal i zero. Since the tracking beam i aligned to different area of the dic, the ignal of the leading beam i delayed 3 µ in order to compare the light intenity on bai of the ame pit. The ame ituation can occur when the mainbeam i between two track. In order to ditinguih between thi cae and the cae where the main beam i on track, the error ignal i divided by the um DS DS of the two atellite diode ignal, er =. When the mainbeam i between two DS + DS track, the um ignal of D S and D S i larger than when the mainbeam i on track o by the ue of thi factor the two ituation can be eparated..4 The ervo mechanim in the CD-player To make ure that the laer i alway focued on the right track everal ervomechanim are neceary. The CD-player ue four actuator to maintain the accurate data readout: Dic motor ervo: The data from the dic mut be read at a contant rate to feed the digital ignal proceing circuit with data. Without control of the dic motor peed, the peed with which the data bit would arrive would depend on the poition of the laer pickup. The dic motor i contantly adjuted o the pickup read the data with contant velocity to keep the data buffer alway half full. Thi mean that the motor peed mut vary between. and.4 m/. Tracking ervo mechanim: In order to enure accurate tracking of the laer beam along the.5 µm track, a tracking ervo i neceary. The ervo mut be able to cope with dic eccentricity, fingerprint on the CD, cratche, vibration of the player, etc. The track may have a maximum ide-to-ide wing of ±.6 mm which i equivalent to 75 track. For thi reaon the radial actuator mut be able to compenate and move at high peed with an accuracy of ±. mm. Thi i done by the ue of an electro motor that conit of a combination of coil and permanent magnet (figure.9). Focu Servo mechanim: Since no dic i perfectly flat, the pecification allow for a vertical deflection ± 6 µm. The laer beam mut tay focued within a ± µm tolerance, otherwie the phae interference between the direct and the reflected light i lot along with the audio data and the tracking information. It i evident that a ervo ytem i neceary to give correct focu of the optical pickup. Thi i alo done by the ue of an electro motor (figure.9). Sledge ervo mechanim: Since the radial actuator ha to move along the complete dic, jut an electro motor doe not have enough range. To make ure that the complete dic can be read, an 9

10 extra actuator i neceary: the ledge ervo mechanim. Thi i a ytem which conit of a DC motor, a worm gear and a reduction gear aembly. In the ledge the radial and focuing actuator are mounted. Becaue uch a ytem exhibit backlah and friction, it i difficult to obtain a mooth and high-peed movement. Therefore thi actuator i only ued for low and large correction in the horizontal direction. figure.9: Actuator for radial poitioning and focuing In order to compare control trategie for the ledge controller a model mut be made from the influence of the ledge movement on the radial actuator from the optical pickup unit. In the next chapter thi model will be derived.

11 3. Modelling of radial and ledge ytem The radial actuator i connected to the ledge by mean of platic arm o the influence of the movement of the ledge i regarded a a diturbance on the radial actuator. Therefore, model of both the radial actuator and the ledge ytem have to be derived. In the next ection a model of the radial actuator will be preented. 3. Model of radial actuator: In order to decribe the influence of the ledge ytem on the radial loop, a model of thi radial loop hould be derived. The radial actuator conit of a coil and a permanent magnet; it can be regarded a an energy converter, which convert electrical current to mechanical motion. The actuator conit in thi way of an electrical and a mechanical part. 3.. Electrical part of radial actuator The electrical part conit of a current-carrying coil which interact with a permanent magnet to create a force field which reult in movement of the optical pickup. The electrical diagram of the actuator i hown in figure 3.. figure 3.: Electrical diagram of radial actuator In thi diagram, L i the inductance of the coil, R i the reitance of the coil and r m i a hunt for current meaurement. The AC-enitivity i repreented by Bl according to Faraday law: dφ B dx emf = = Bl (3.) dt dt and x i the diplacement of the len. Now Kirchhoff law can be applied: d d u( t) = ( R + rm ) i( t) + Bl x + L i( t) (3.) dt dt To determine whether or not the inductance i important in the actuator, the break frequency of the low-pa filter, formed by the two reitance and the inductance, can be computed: I( ) L U ( ) = = R + r L R r m + + (3.3) m ( + ) L In the data heet from B&O the value can be found (appendix A.). The break frequency i computed to be:

12 R + rm 8 + fbreak = = 8. 3 khz 6 L 65 = (3.4) π π Since the frequency range of the total ytem i only - khz, the inductance can be neglected. When a voltage u i applied to the coil a current i will flow through it and due to the magnetic field from the magnet a force f el proportional to the current will be generated according to Laplace law: d F = Idl B (3.5) Uing thi equation with equation 3., the next expreion for the electrical force can be derived: d Bl u ( Bl) x f dt el ( t) = Bl i( t) = (3.6) R + r m 3.. Mechanical part of radial actuator Now the mechanical part of the actuator ha to be modeled. The optical pickup unit i placed in the ledge by four lever arm which hold the pickup in the middle of the operating range. Thi ytem can be divided into two part, a ma from the len and a ma from the lever, ee figure 3.. f el k, b k, b figure 3.: Actuator with relevant parameter Thi approach lead to a fourth order model of the actuator but ince it i obviou from the bode diagram upplied by B&O that the dominating dynamic can be decribed by a econd order tranfer function, the actuator will be regarded a one ma which i attached to the ledge. Since the ma of the ledge i approximately 4 gram and the ma of the optical pickup i only,56 gram the influence from the pickup on the ledge i neglected. In thi cae the ledge can be regarded a a wall with a certain movement and the following chematic repreentation of the ytem can be made: pickup ledge k f el m c x v figure 3.3: Schematic diagram of actuator and ledge

13 The following equation of motion can now be derived: mɺx = f el k( x v) c( xɺ vɺ ) (3.7) If equation 3.6 and 3.7 are combined the next equation i formed: Bl u ( Bl) mx ɺ = xɺ kx cxɺ + kv + cvɺ (3.8) R + r R + r m m The tate are choen to be x = [ x x] T, the diturbance i choen to be v [ v v] T ɺ = ɺ and the input to thi ytem i u. With thee variable a tate pace repreentation of the econd order ytem can be formed: xɺ = k c ( Bl) x + Bl u + k c v m m m( R r m( R r m ) m ) + m m + (3.9) y = [ ]x Mot of the variable can be found in the dataheet of B&O, the pring and damper contant are the only unknown variable. They can be found by fitting the parameter to a meaured bode plot of the ytem. In [] thi ha been done. All value can be found in appendix A.. The bode diagram of thi ytem i depicted in figure 3.4: -6 Gain [db] Frequency [Hz] Phae [deg] Frequency [Hz] figure 3.4: Bode diagram of radial actuator 3

14 3. Model of ledge ytem The ledge ytem conit baically of three part: DC-motor Gear mechanim Moving ma in which the optical pickup i mounted: the ledge itelf In the next ection all thee part will be modeled. 3.. DC-motor The DC-motor conit of an electrical part and a mechanical part, ee figure 3.5. In the next ubection thee part are decribed Electrical part: The electrical part of the motor conit of the voltage ource (V), a reitor (R), an inductance (I) and a back (induced) electromotive force (emf) generated by the turning of the motor. The torque delivered by the motor i proportional (with the armature contant k m ) to the current that flow through the circuit. emf T r figure 3.5: Model of DC-motor For thi ytem the next equation hold: di V emf = L + Ri, dt emf = k θɺ o emf ; di R k = i( t) emf θɺ + V (3.) dt L L L and: T = kmi (3.) In thi ytem, the time contant of the integrator can be computed to determine if the inductance play an important role in the total ytem. The time contant equal: R τ = (3.) πl The ued value lead to a time contant of,6. -3 econd. Thi i equal to 637 Hz and thi i beyond the bandwidth of the ledge ytem. However, the abence of the induction lead to an algebraic loop when the model ha to be olved o it will be ued in the model. 4

15 3... Mechanical part In the mechanical part friction i preent o three cae have to be ditinguihed, namely the cae when there i movement, the cae when there i no movement and the driving torque i le then the tatic friction and finally the cae where there i no movement but the driving torque i bigger than the tatic friction. Now the mechanical part can be decribed by the following equation: if v ( T T ɺ θ ( ɺ r b Tcmign θ )) J m if v = and ( T T ɺɺ θ = r ) < Tcm (3.3) ( T T T ign( ɺ r cm θ )) if v = and ( T T r ) > Tcm J m Here J m i the moment of inertia of the motor, k m the armature contant of the motor (equal to k emf ), b the motor damping coefficient, T r the torque reduced from the load and T cm the motor tarting torque. Equation (3.) and (3.) together decribe the behavior of the motor. 3.3 Gear tranmiion A known feature in the ledge ytem i backlah. In order to decribe thi phenomenon the gear mechanim hould be treated a a eparate part of the model. Thi part hould make the connection between the DC-motor and the ledge itelf. The gear mechanim conit of a worm wheel (), two cogwheel ( & 3) and a toothed bar (4), ee figure 3.6. DCmotor 3 4 ledge figure 3.6: Gear mechanim of ledge The backlah i mot viible between the two cogwheel ( & 3) o the gear mechanim can be plit in two part, one part that conit of the worm wheel and one cogwheel and one part that conit of the econd cogwheel and the toothed bar. Between thee part a dead zone can be implemented to imulate backlah. To determine the gear ratio, the teeth on the wheel and the toothed bar can be counted and for the computation of the ratio of the worm wheel the dataheet can be ued. Thi i done in appendix A.. With the backlah, the torque delivered to the load (T l ) can be written a: Tl =, if θ mr θ l θ b (3.4) T = k ( θ θ θ ) ign( θ θ ), otherwie l mr l b mr l 5

16 where k i the gear tiffne, θ mr and θ l the motor poition reduced to load and the load poition repectively and θ b the amount of backlah. The torque on the motor reduced from the load can be written a: T = (3.5) r NT l 3.4 The ledge The torque delivered by the firt part of the gear mechanim i directly fed to the econd part on which the ledge i mounted. In thi ytem, friction play an important role. There are four kind of friction that are incorporated in thi ytem: Coulomb friction: Thi friction depend only on the ign of the velocity. It can be decribed by equation (3.6). F = FC ign(v) (3.6) Vicou friction: Thi part of the friction depend on the value of the velocity and i decribed by equation (3.7). F = f v (3.7) Stiction: Experiment have hown that tiction play an important role in the ledge mechanim. Thi tiction i poition dependent and it i modeled by the next equation: F = F + F in( 5v) (3.8) Stribeck friction: In general, the friction force due to tiction doe not drop uddenly when velocity increae. Stribeck oberved that the friction ha an exponential hape with a minimum at the Stribeck velocity. The total friction force can now be decribed with a combination of all friction part. Thi done in equation (3.9). Fv ( v) if v F = T if v = and T < F (3.9) F ign( Fe ) if v = and T > F Where: b ɺ θ Fv = FC ign( ɺ θ ) + f ɺ θ + ( F Fc ) e Thi lead to friction force depending on the velocity a een in figure

17 Friction force Velocity Figure 3.7: Friction model for ledge In the real ytem the ledge doe not alway move when a pule i given. To model thi phenomenon the tiction can obtain different value. If the tate are choen a x = [ i ] T m θ ɺ m θ m θ ɺ l θ l and the input a u = [ V ] a tate pace decription of the ledge ytem can be formed: ( R x() kemf x(3) + u L x(3) xɺ = ( k m x() k ( x() x(4) θ b )gn( x() x(4)) Tcm gn( x(3)) b x(3) J m n n n x(5) b x ( k ( x() x(4) θ b )gn( x() x(4)) FC ign( ɺ θ ) f ɺ θ ( F Fc ) e J n n n y = [x(6)] Since thi i a highly non linear model the bet olution to do imulation i to make a model in Simulink. Mot of the modeling in Simulink i traightforward. However, there are ome poible problem which hould be avoided uch a numerical problem, unwanted zero-croing, algebraic loop, etc. In figure 3.8, the global model of the ledge ytem i depicted. The variou part of thi model can be found in appendix B. (5) 7

18 input voltage U torque torque input poition motor velocity motor electrical part motor torque ledge on motor mechanical part motor velocity motor velocity ledge -Kgear ratio velocity ledge toque to ledge torque to ledge poition ledge poition ledge -Kgear ratio poition ledge ledge gear mechanical part of the ledge ytemytem figure 3.8: Global Simulink model of the ledge ytem Thi ytem can now be coupled to the radial actuator which i alo preented in appendix B. The coupled Simulink model of the ledge ytem and the radial actuator with controller i given in figure 3.9. input voltage to ledge ytem input v oltage poition ledge v elocity ledge x v x eccentricity radial controller ledge ytem ur error control ignal radial actuator Ramp ramp Figure 3.9: Coupled Simulink model of ledge ytem and radial actuator In the next chapter controller for the radial loop and the ledge ytem will be made and the influence of the ledge movement on the radial loop will be invetigated. 8

19 4. Deign of radial controller and pule teered ledge controller In thi chapter, a controller for the radial loop will be computed and the exiting control olution to the ledge will be imulated. 4. Controller for the radial loop A tated in chapter, the radial controller hould be able to follow the track on the CD. To uppre diturbance like hock thi ytem hould have a high bandwidth to compenate quickly for thee. However, to uppre fault like cratche on the CD the controller hould have a low bandwidth, otherwie the radial actuator will ee the cratch a a track and will tart to follow it. A trade-off ha to be made between thee two ituation. In addition, when the radial controller ha a higher bandwidth, acoutic noie i more audible when playing compact dic. A commonly ued controller i a PID controller with a bandwidth of Hz. Baed on the model of the radial actuator derived in the previou chapter uch a controller i deigned traightforward. The controller i depicted in figure Gain [db] Frequency [Hz] 4 Phae [deg] Frequency [Hz] Figure 4.: Bode diagram of the controller of the radial loop The radial control ignal cannot grow boundle while compenating o a aturation on thi ignal ha to be implemented in the model. When the ignal aturate, the controller till think the computed ignal i fed to the ytem and accordingly the integral action will tart to wind up and thu the control ignal will grow more. Thi lead to unneceary big error in the radial loop. To avoid thi phenomenon, an anti-windup ytem i added. Thi i done by ubtracting the difference between the computed and aturated control ignal from the radial error that goe to the integrator. See for thi implementation appendix C, figure C.. In figure C. in the ame appendix the Nyquit diagram of the open loop radial ytem i depicted. 9

20 4. Pule teered controller for the ledge ytem A tated before, the ledge i only ued for low and large movement in the horizontal direction. There are everal poible olution to control the ledge. At the moment, the ledge i controlled by ending pule to the DC-motor when the ledge ha to be moved. Thi method i choen ince it i difficult to make a continuou controller due to friction, tiction and backlah in the ytem. A problem with thi method i that it i not certain that the ledge will move when a pule i given. If for example the tiction i to high or the backlah i to big it i poible that the ledge doe not move when a pule i given. Thi i the reaon why the pule have an increaing amplitude. The diadvantage of thi way i that there i no way to determine how big the amplitude of the pule ha to be. It i poible that the firt pule i jut a bit to low and that the econd i much too big. In thi way a bigger diturbance than needed will be impoed on the radial loop. It i even poible that the pule bring the radial actuator to the other ide of it non-linear area. Thi can lead to mute or lo of the track. Another problem which occur i that, ince the ledge doe not move at the deired moment, the radial loop can aturate which lead to bigger error ignal. Thi control method can be implemented in Simulink through the ue of a S-function that generate the pule. The ledge i uppoed to move when the radial loop get into it non-linear area. The pule are therefore triggered by a threhold on the control ignal of the radial loop. Since it i not known when the ledge i going to move, thi threhold hould be lower than the aturation voltage of the radial loop. The aturation voltage i et to ± 3 V. The threhold i then et to,75 V. In the imulation the balance between the ramp part of the radial control ignal and the inuoid part (caued by eccentricity of the dic) i exaggerated. Thi i done to make ure the ledge move in the imulated time. The model of the controlled ytem i given in figure 4.. poition ledge input voltage velocity ledge x v x eccentricity ledge ytem ur error ledgecontrol radial ytem ramp radial controller control ignal to radial actuator poition of ledge control ignal to ledge ledge controller figure 4.: Pule controlled ytem A tated earlier, the purpoe of the ledge control mechanim i to avoid the radial loop from getting into aturation. The control ignal to the ledge and the radial actuator look now a follow:

21 Voltage.8 Voltage time [] time [] figure 4.3: Control ignal to ledge figure 4.4: Radial control ignal The radial error ignal with and without aturation of the radial loop look than a depicted in figure 4.5 and 4.6. x -7.5 x error Error time [] Time figure 4.5: Radial error ignal figure 4.6: Radial error ignal without aturation with aturation The diturbance impoed by the ledge are not that big but if the radial actuator aturate, the influence i obviou. To avoid thi, a continuou controller can be ued. Thi i howed in the next chapter.

22 5. The deign of a continuou controller A controller baed on pule give a large diturbance on the radial loop. Therefore, it i better to make a more mooth control for the ledge in order to minimize diturbance on the radial loop and to avoid aturation. Thi ytem exhibit a lot of friction and backlah and thi impoe retriction on the ledge controller. If there i overhoot in the ytem for example, the controller will try to compenate for thi by giving a control ignal of oppoite ign. However, the backlah will generate a torque to the ledge which i too big and there will be overhoot in the oppoite direction. Therefore, aturation ha to be implemented uch that the control ignal can be only poitive or negative during a movement. Another problem i the tiction, the ledge ha to move all the time otherwie it will tick and till a non-mooth movement will be the conequence. 5. Control trategy for ledge To control the ledge, a PD controller i deigned ince perfect tracking i not neceary. Becaue it i not recommended to have more movement in the total ytem then i neceary, the ledge i moved emi-continuouly. Thi mean that the ledge i only moved at time when the radial control ignal i about to aturate but when it i moved, the ledge will follow a continuou reference. To limit the time the ledge ha to be moved, thi reference will bring the radial actuator from one bound of it range to the other. 5. Sytem and controller In the end of chapter three the tate-pace decription of the ledge ytem ha been derived. In order to deign a PD controller, thi ytem ha to be linearized. Thi lead to the following ytem: R L k m xɺ = J m y = k J mn k J n n kemf L b J m x k J mn k J L x + u f J (5.) The bode diagram of thi linearized ytem and the deigned controller can be een in figure 5. and 5..

23 Gain [db] Frequency [Hz] Gain [db] Frequency [Hz] -5-5 Phae [deg] -5-3 Phae [deg] Frequency [Hz] 3 Frequency [Hz] figure 5.: Bode diagram of figure 5.: Controller linearized ytem Every time the radial control ignal reache a certain level, a S-function generate a 3 rd order mooth reference ignal which bring the ledge to the deired poition. A mentioned earlier, the ytem exhibit tiction and backlah. Thee two phenomena give rie to a trade-off in bandwidth of the controller. Becaue of the backlah, the control ignal i not uppoed to change ign ince thi will lead to a limit cycle. Overhoot in the ytem hould be avoided becaue then the ledge will top and tiction will prevent it from moving until the control ignal ha enough energy to move it again and overhoot occur again, thi lead to non-mooth movement. The bandwidth of the controller hould not be too high in order not to have overhoot becaue of the backlah (ee appendix C, figure C.5) but it hould not be too low a well ince a low bandwidth implie that it take a long time before the control ignal ha enough energy to overcome the tiction and a tep like movement occur which lead to big diturbance on the radial loop (ee appendix C, figure C.6). To overcome a part of the backlah a feed forward ignal i implemented a well. Thi ignal conit of three pule which are too low in amplitude to move the ledge but high enough to move the DC-motor and gear mechanim. Thi feed forward i fed to the ytem jut before the reference ignal. The Simulink model of the controlled ytem with feed forward can be found in appendix C, figure C.3. In figure C.4 the Nyquit diagram of the controlled ytem i depicted. Since the ledge i moved in a mooth way (ee figure 5.3), the radial error i overall maller than in the cae the pule were ued a can be een in figure 5.4. If figure 5.4 and 4.5 are compared, note that the imulation time in figure 5.4 i twice the time from figure 4.5. Thi way of controlling the ledge give better reult than the pule control approach. The ledge i not moved a often a i the cae when pule are ued, the error i maller and aturation of the radial loop can be avoided. There i only one problem: in the real ytem the poition of the ledge i not known ince it i not meaured. To overcome thi problem another ignal ha to be ued a feedback to the ledge controller. Thi i done in the next paragraph. 3

24 3.5 x -3 Sledge poition Reference x Sledge diplacement.5 Error Time Time figure 5.3: Sledge diplacement figure 5.4: Radial error 5.3 Uing the radial control ignal Since the actual poition of the ledge i not known, another ignal ha to be ued to control thi poition. When the ledge i moved, the radial control ignal will drop. Thi ignal can therefore be ued to control the ledge. Another advantage of chooing thi ignal i the fact that the ledge i moved in the firt place to avoid that the radial controller aturate. Thi aturation depend on the radial control ignal o it would be bet to control thi ignal Reference ignal The radial control ignal conit of generally two part. The firt part i a ramp caued by the fact that the data on the dic i tored in piral track and the econd part i a inuoid caued by the eccentricity of the dic. Only the ramp i important conidering the ledge movement and thu the radial control ignal ha to be filtered. A low-pa filter however give too much delay o another way ha to be found. In thi report a notch filter i ued that filter out the inuoid at the rotational frequency. Thi frequency i not the ame throughout all poition on the dic but varie from inner to outer diameter from 9 till 4 Hz. Thi filter can therefore not be ued in the real ytem, however, there are way to plit the radial control ignal in it variou frequency content and thu thi control trategy can be implemented. The ledge hould move in a mooth way to keep the diturbance on the radial loop a mall a poible o alo a mooth reference ignal hould be ued. In figure 5.5 a poible reference ignal i depicted. 4

25 figure 5.5: Reference ignal Every time the radial control ignal reache a threhold, the deired trajectory for thi ignal i fed to the controller. Thi done by implementing a S-function in Simulink which generate thi reference on the deired moment Sytem and controller To be able to deign a controller a model hould be made for the ytem with a input a voltage to the ledge and a output the control ignal to the radial actuator. Thi i done by linearizing the non-linear ytem in Simulink. With thi model, a controller can be derived. The linearized ytem and the deigned PD controller with notch can be found in figure 5.6 and 5.7. Gain [db] Gain [db] Frequency [Hz] 3 Frequency [Hz] 5 5 Phae [deg] Phae [deg] Frequency [Hz] 3 Frequency [Hz] figure 5.6: Linearized ledge ytem figure 5.7: Sledge controller A can be een in figure 5.6, there i +8º extra phae in the ytem. Thi can be explained by the fact that when the ledge move forward, the radial control ignal move downward. A minu ign ha therefore been introduced. The controlled ytem ha a bandwidth of 45 Hz. The Nyquit diagram can be found in appendix C, figure C.7. The total control cheme i depicted in appendix C, figure C.8. 5

26 5.3.3 Reult If the correct bandwidth i ued together with the feed forward a mooth movement reult, ee figure 5.8. In the next figure the reference and actual filtered control ignal are depicted x -3.5 Filtered radial control ignal Reference Diplacement.5 Voltage Time Time figure 5.8: Sledge movement figure 5.9: Reference following The radial control ignal and the radial error ignal are given in figure 5. and 5. (compare to figure 4.5 and 5.4). The ledge control ignal can be found in appendix C, figure C.9. 4 x Voltage Error Time Time figure 5.: Radial control ignal figure 5.: Radial error A can be een from figure 5. the radial error caued by the ledge movement i not o obviou than when pule are ued. When the ledge i moved continuouly, movement 6

27 can be predicted better and the ledge doe not have to move a many time a wa the cae with pule teering. A ignificant error can be een when the ledge tart to move, thi error can be made maller by uing a radial controller with a higher bandwidth at the moment the ledge tart to move. Thi i done in the next paragraph. 5.4 Gain cheduling Since the radial error i mot obviou when the ledge tart to move, it i poible to ue a controller with a higher bandwidth for the radial loop during a hort time when the ledge move. Thi approach i alo poible with pule teering, however, if a controller with higher bandwidth i ued, it i poible that the actuator will follow a cratch on the CD. In the cae of pule teering the bandwidth hould be increaed every time a pule i given ince there i no telling when the ledge will move, thi mean that the bandwidth ha to be increaed too many time. With the emi-continuou moving thi approach i poible becaue the ledge jut move a few time. A can be een in figure 5., the radial error i mot obviou when the ledge tart to move o the bandwidth only ha to be increaed in the beginning of the reference following. Since the ledge doe not move a many time a i the cae with pule teering, the gain cheduling method can be applied. In order to have a mooth tranfer from one controller to the other ome precaution have to be taken. If the two controller are jut witched, a tep in the radial control ignal will occur. To prevent thi, the tate of both controller have to be the ame when the witching i done. Thi can be done by uing the ame integrator in the Simulink model and jut witch the other controller part. In order to do o, the controller hould be divided in part that make it poible to et the integrator free. The radial controller ued i a PID controller, depicted in figure 4.. The tranfer function of the total controller i: Y p p τ τ = gain U + + i (5.) z p τ τ τ + Thi tranfer function can be plit in the three part, converted to tate-pace decription and to block diagram: Proportional: Y τ u p τ y p = gain τ gain z U τ integral: z Y U = τ p gain τ z τ i τ p xɺ = gain τ z y = x u u τ τ p z gain y 7

28 Derivative: gain Y U = gain τ d τ + d gain u xɺ = x τ d τ d y = x + gain u u gain τ d + + y τ d The total block cheme of the controller with the gain cheduling and anti-windup mechanim implemented can be found in figure C., appendix C. The radial error with the gain cheduling implemented i depicted in figure 5., a cloe up on the chedule time i given in figure 5.3. x -7 x -8 8 Error Error Time Time Figure 5.: Error with figure5.3: Cloe-up on gain cheduling radial error If figure 5. i compared to 5., it i clear that the bigget error are reduced a wa the idea with the gain cheduling. Thi control trategy provide a good way to move the ledge without ue of extra meaurement and to keep the radial error mall without uing a high bandwidth for a long time. 8

29 6. Concluion and Recommendation A model ha been developed for the ledge ytem coniting of a DC-motor, a gear mechanim and a ma to which the optical pickup ytem i attached. Simulation have been done on the nowaday ued control method baed on pule. Thee imulation how that the pule give many diturbance on the radial tracking ytem. Since it i not certain when the ledge move due to backlah, tiction and friction pule of increaing amplitude have to be ued. A better approach i a continuou movement of the ledge which bring the radial actuator from one end of it range to the other. A PD controller with poition give good reult in imulation but unfortunately thi ignal i not meaured and can therefore not be ued. If intead the filtered radial control ignal i ued for feedback imulation how alo good reult. The ledge doe not have to move a often a wa the cae with pule teering and the radial actuator capable of compenating the continuou movement of the ledge. The tiction give rie to the problem that the reference i not followed immediately but jut after a hort period. Thi lead to the fact that there i a tep wie tart of the ledge movement and the radial error i mot obviou at thi moment. To decreae thi error, a gain cheduled radial controller ha been implemented that double the bandwidth at the moment the ledge tart to move. Thi total ytem give promiing imulation reult. Due to the fact that the teting ytem wa not finihed at the end of thi training period, real time experiment could not be done. The parameter for the ledge ytem uch a friction parameter, tiction, backlah and motor contant are not determined experimentally o reearch to thi data hould till be done. It i alo recommended to invetigate the poibilitie of filtering the radial control ignal in uch a way that only the ramp will be viible without too much delay. Another reference ignal could be deigned which incorporate the fact that there i backlah and tiction preent in the ytem. The feed forward ignal can be expanded to compenate for tiction and friction a well, poibly with iterative learning each time the ledge i moved ince alway the ame reference i ued. 9

30 Reference [] Ramu Søndergaard Anderen, Karten Gjørup Hanen & Kritian Bøcher Poulen, Optical control: A control ytem for compact dic player, Aalborg Univerity, [] Kawa A. Abdulrahman, Ulineær model af DC-motor, Aalborg Univerity, 993 [3] Brian Anderon & Ragnar Viktor Karlon, Regulering af optik pickup i CDafpiller, Aalborg Univerity, [4] Han Chritian Schøien, Jeper Ramuen & Enrique Vidal Sanchez, Focu control of optical pickup in CD-player, Aalborg Univerity, 998 [5] Karl J. Ǻtröm, Control of ytem with friction, Lund Intitute of Technology, 985 [6] Mikael Sternad & Stefan Rönnbäck, A frequency domain approach to Anti- Windup compenator deign, Uppala Univerity, 993 3

31 Movingma(m)- Q-factorcoil Voltageoncoil, Kg Reonancefrequency4 8 - ACenitivity(Bl). 3 Vdc DCenitivity Hz Reitance(R),5.3 N/A Shunt(rm),4.-3 m/v Inductance(L) c W k,65.-4,4 HN/m Forthecomputationofthepringandampercontant, - 53, - N/m convertedtoatranferfunctionwiththenextformula: thetatepacedecriptioncanbe () Sothetranferfunctioni: Appendix A: () fromtheekandcanbederiveda: (4) (3) tatedintheabovetable. fitedtothebodediagramoftheradial Nowtartingvalueoftheeparameterareavailableandin[]theevaluehaveben ytemtodeterminethecorectvalueaare Radial actuator: Value radial actuator and ledge ytem Parameter Min. Typ. Max. Unit 3

32 Parameter ledge ytem: Parameter Value Unit Motor inertia (J m ),. -4 Kg. m /rad Inductance of motor (L) 4,. -5 H Reitance of motor (R),6 Ω Armature gain (k m ) 4,88. - Nm/A Emf gain (k emf ) 4,88. - V/rad. - Motor damping (b) 7,6. -5 Nm/rad Motor tarting torque (T cm ), Nm/rad A. Computation of the gear ratio The gear mechanim conit of a worm wheel (), two cogwheel ( & 3) and a toothed bar (4), ee figure A... DCmotor 3 4 ledge Figure A..: Gear mechanim of ledge Data Toothed bar (4) Length: 4 mm Teeth: 4 Cogwheel (3) Teeth big wheel: 68 Teeth mall wheel: 5 Cogwheel () Teeth: 7 In the dataheet from B&O the total gear ratio from DC-motor to tranlation of the ledge can be found: krad/m. The ratio from the motor to wheel () can be derived by dividing the total ratio by the ratio from cogwheel () to the tranlation of the ledge, which can be determined from the above data: tranlation ledge = = 5, rad. motor worm ratio 68 π ratio worm wheel = 5,9683 3

33 Appendix B: Simulink model of the ledge ytem In thi appendix the variou part from the ledge model a een in figure 3.8 are preented. Electrical part DC-motor: -Kemf gain velocity motor -Kreitance U -Kinductance Saturation current i -Karmature contant torque torque Mechanical part DC-motor: velocity motor torque torque input -Kinertia motor thetam tmot v motn v mot zero croing detector poition motor Friction f riction v mot Tormot Friction block: torque ledge on motor Tcm fm u tanh(*u) -C- vmot u NOT friction Tcm min Tormot u tanh(*u) 33

34 Zero croing detector: vmot NOT vmotn tmot u <= Tcm Thi part i to make ure the velocity doe not change ign in one integration tep becaue of numerical iue. The velocity at the preent integration tep i compared to the velocity at the previou one by mean of a memory block and an additional very mall delay to make ure the change of ign i detected. If there i no change, the new velocity equal the incoming velocity. If there i a change and the abolute value of the torque delivered to the motor i maller than the coulomb friction to the motor, the new velocity i et to zero. Otherwie, the ign of the velocity will change and the coulomb friction will change ign o the torque i again in oppoite direction of the velocity. Thi lead again to a change of ign of the velocity and of the coulomb friction and in thi way to a velocity ignal that i changing ign every integration tep while in the real ytem thi i not poible. If the torque delivered to the motor i greater than the coulomb friction the change of ign i a real change and the incoming velocity i paed through. Gear: torque ledge to motor poition motor gear ratio -Kgear ratio torque to ledge Kg gear tiffne Backlah poition ledge 34

35 Mechanical part of ledge: tiction generator poition ledge tiction T v f riction tiction Friction poition ledge velocity ledge Integrator retriction on change of velocity trehold on velocty v l v v el out v el in v ln tled F trehold on torque inertia ledge torqueout torquein toque to ledge Friction: Fc f u tanh(*u) Fc F u exp(-b*u) tanh(*u) -C- NOT u friction v 3 tiction min T u Retriction on velocity: vl NOT vln tled u <= 3 F 35

36 Stiction generator: poition ledge F*in(5*u) F tiction Simulink model of radial actuator: Kr x Cr v 3 ur /(R+rm) Blr B*l -Kma actuator x x Cr damper Kr pring Blr induction 36

37 Appendix C: Controller Radial loop: taudp. taudp.+ derivative gain loop radial control ignal error tauint. integral proportional antiwindup gain figure C.: antiwindup Radial controller Nyquit Diagram Imaginary Axi Real Axi figure C.: Nyquit diagram controlled radial ytem Sledge ytem: feed forward radial control ignal f eed forward reference em ledge controller eccentricity error ledgecontrol poition ledge input voltage velocity ledge ledge ytem x v ur x error control ignal radial ytem ramp radial controller figure C.3: PD controlled ledge ytem with poition feedback 37

38 Nyquit Diagram 3 Imaginary Axi Real Axi figure C.4: Nyquit diagram PD controlled ledge ytem with poition feedback x Diplacement.5 Diplacement Time Time figure C.5: Too high bandwidth figure C.6: Too low bandwidth Nyquit Diagram.5.5 Imaginary Axi Real Axi Figure C.7: Nyquit diagram of controlled ledge ytem with radial control ignal a feedback 38

39 radial control ignal f eed f orward reference em error ledgecontrol input voltage poition ledge velocity ledge x v x numufilt() denufilt() ur error control ignal figure C.8: Control cheme Sledge control ignal Time figure C.9: Control ignal to ledge (pule are viible before continuou control tart) 39

40 error control ignal chedule ignal aniwindup figure C.: Gain cheduled controller with anti-windup 4