Shaped Time-Optimal Control for Disk Drive Systems with Back EMF, Slew Rate Limits, and Different Acceleration and Deceleration Rates

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1 Shped ime-optiml Control for Disk Drive Systems with Bck EMF, Slew Rte Limits, nd Different Accelertion nd Decelertion Rtes Chnt L-orpchrpn nd Lucy Y. Po Deprtment of Electricl nd Computer Engineering niversity of Colordo, Boulder, CO nd Abstrct his pper derives shped time-optiml servomechnism (SOS) controllers for flexible structures consisting of second-order rigid body with dmping nd dmped flexible mode. he pproch uses input shping to ccount for the dmped flexible mode by ltering the time-optiml velocity profile tht moves the second-order rigid body with dmping from rest to rest. Considering the slew rte limit constrints, the mneuver is ctegorized into three rnges due to their different shped input profiles. For one of the mneuver rnges, when systems hve different ccelertion nd decelertion cpbilities, the time-optiml control for the second-order with dmping rigid body portion hs to be solved numericlly. However, the SOS control for this mneuver rnge cn still be derived nlyticlly by using the curve-fit pproximtion of the switching time of the time-optiml control. Simultions show tht the derived SOS control lws yield ner timeoptiml performnce without unwnted residul vibrtion t the end of the mneuver.. Introduction It is desirble to hve servomechnisms, e.g., disk drive systems, effect minimum time response to set point chnges. o reduce the time response, the system mss is often decresed by using lighter mteril, but this cn led to the structures becoming quite flexible. Rpid nd precise control of flexible structures hs proven to be chllenging nd rich re of reserch [, 5]. While time-optiml feedbck control lws tht simultneously ccount for the motions of the rigid body nd one flexible mode hve been derived [, ], these methods re not esily generlizble for considering dmping, different ccelertion nd decelertion rtes, slew rte limits, or dditionl flexible modes. Phse-pe pproches hve been used to derive ner time-optiml feedbck controllers such s the proximte time-optiml servomechnism (POS) nd extended POS (XPOS) [3,,, 3, 5]. POS does not explicitly ccount for flexible dynmics while XPOS ccounts for flexible dynmics only in the finl settling phse of the his work ws supported in prt by hi Government Scholrship, the S Ntionl Science Foundtion (Grnt CMS-459), the Colordo Center for Informtion Storge, nd niversity of Colordo Fculty Fellowship. x ref xe Σ y y Controller Flexible Structure Fig. : Closed-loop system with the SOS control. Pos controller, nd neither of them ccounts for the slew rte limits. Input shping is used to ccount for the flexible dynmics during the entire slewing motion in [4], but the design procedure requires the feedforwrd control commnd to be numericlly re-computed for ech set point chnge. We recently developed shped time-optiml servomechnism (SOS) control technique bsed on shping timeoptiml phse-pe trjectories [6-9]. A block digrm of closed-loop system with SOS controller is shown in Fig.. he SOS pproch considers the rigid nd flexible dynmics seprtely. It strts with the time-optiml control tht moves the rigid body portion of the system from rest to rest nd then shpes this control with n input shper to ccount for the unwnted flexible mode. he resulting shped time-optiml control leds to n ltered phse-pe velocity profile such tht residul vibrtion is eliminted. he SOS pproch is more mthemticlly trctble nd is generlizble for considering dmping, different ccelertion nd decelertion rtes, slew rte limits, or dditionl flexible modes. Compred to feedforwrd input shping, the SOS control lws utomticlly hndle some set point chnges without hving to re-compute the shped feedforwrd commnd. While yielding ner time-optiml performnce, the SOS pproch cn ddress flexible modes tht cn not be ddressed in the POS or XPOS. Further, becuse the SOS control lws use phse-pe similr to those used in POS or XPOS, implementing the SOS control lws in pplictions tht use controllers similr to the POS or XPOS controllers should not need significnt re-configurtions. In some pplictions, the ccelertion nd decelertion cpbilities of the systems re not equl. In this pper, we extend the SOS control lws developed in [7, 8] for flexible el

2 volt + R cur K b K t J K f s ζω n s ω n s s + pos Fig. : Block digrm of single mode flexible structure driven by voice coil motor. R, K t, J, nd K b re coil resistnce, torque constnt, moment of inerti, nd bck EMF gin, respectively. structures hving second-order rigid body with dmping nd dmped flexible mode to further ccount for different ccelertion nd decelertion cpbilities. Fig. shows block digrm of disk drive system driven by voice coil motor nd considering the effect of the bck electromotive force (bck EMF) which cts like viscous friction on the system. he closed-form solutions of the time-optiml control for the second-order rigid body with dmping system with different ccelertion nd decelertion cpbilities nd for the thirdorder rigid body system s described in [9] re not known. he time-optiml controls must be solved numericlly. However, we still cn derive the SOS control nlyticlly by using the curve-fit pproximtion of the switching time of the timeoptiml control for the system hving the second-order rigid body with dmping nd the curve-fit pproximtions of the switching curves for the system hving the third-order rigid body [9]. his pper is orgnized s follows. We derive the SOS control lw for flexible structures tht consist of secondorder rigid body with dmping nd dmped flexible mode considering slew rte limits nd different ccelertion nd decelertion rtes in Section. Section 3 presents simultion results using the SOS control lws derived in Section. Finlly, concluding remrks re given in Section 4.. Shped Phse-Pe Feedbck Control A servomechnism the non-flexible dynmics consist of one integrtor nd one rel pole (representing secondorder rigid body with dmping) nd the flexible dynmics consist of one dominnt flexible mode with dmping cn be modeled using stte spce differentil eqution s follows: ẋ Ax Bu () A ω n ζω n + B b b 4 x x r x f ẋ f he stte x represents displcements, x r nd x f nd velocities, nd ẋ f rigid body nd flexible body re denoted by the subscripts r nd f, respectively. ω n is the nturl frequency nd ζ is the dmping rtio of the flexible mode. is the time constnt of the dmping (or bck EMF) of the rigid body. For the system in Fig., K t K b RJ, b K t RJ, b 4 K f, nd u is voltge control commnd. he trjectory of the sttes x r nd due to n rbitrry constnt input u k t time t cn be described s x r t x r e t u k b t e t t e t u k b e t () x r nd re the sttes x r nd t the initil time, respectively. From (), the x r phse-pe trjectory of the second-order rigid body mode with dmping is s follows: x r x r u k b u k b u k b (3) he shped input is the result of the convolution of the time-optiml control with the two impulses of the input shper tht hve mplitudes of nd [4] M M M M e ζ ζ (4) nd occur t times nd, ω n ζ is the dmped nturl frequency. When considering the slew rte limit constrint, the rnge of mneuvers needs to be ctegorized into three cses becuse they hve different shped input profiles. Fig. 3 shows the unshped nd shped time-optiml input profiles for these three cses. he time-optiml input is either t the mximum control or the minimum control α, α indictes different ccelertion nd decelertion cpbilities. Cse is when the move distnce L is smll enough such tht the shped time-optiml input does not cuse the velocity to rech the slew rte limit. he unshped nd shped time-optiml inputs for Cse re shown in Figs. 3 nd 3b, respectively. Cses nd 3 re when the unshped inputs cuse the velocity to rech the slew rte limit. he move distnce L in Cse is lrger thn in Cse but smller thn in Cse 3. Figs. 3c nd 3e show the unshped timeoptiml inputs for Cse nd Cse 3, respectively. he difference between Cse nd Cse 3 is tht in Cse the time durtion t t of the second control level of the unshped time-optiml input is shorter thn the shper length, s in Cse 3 the time durtion t t of the second control level of the unshped time-optiml input is longer thn the shper length. he shped time-optiml inputs for Cse nd Cse 3 re shown in Figs. 3d nd 3f, respectively. Note tht if! α, then the shped time-optiml inputs for Cse during the time t "$# t t nd for Cse during the time t "%# t t re positive nd vice vers. If α & b, then the shped time-optiml inputs for Cse during the time t "'# t t nd for

3 Cse 3 Cse Cse α b α b α t Cse 3 during the time t " # t t vers. t t () (c) (e) t t t 3 t ime-optiml input = * α Shper α t 3 b α b α b α α b α b α α α t t t t (b) t t (d) t (f) t t t 3 t Shped input Fig. 3: nshped nd shped input profiles for three move distnce rnges,. re positive nd vice.. Cse he time-optiml input with different ccelertion nd decelertion rtes (Fig. 3) tht drives the second-order rigid body with dmping portion of the system from rest to rest is u t t t α t t t (5) he switching time t nd mneuver time t cn be obtined from the following constrint equtions due to the rest-to-rest boundry conditions: α e t t e t α α t αt L b (6) t t 3 t t 3 t 3 Assuming tht the dmped nturl frequency is lrge enough such tht the shper length is smller thn the lengths t nd t t of the pulses of the unshped timeoptiml input, the shped time-optiml input (Fig. 3b) is u t t t t α t α t α t t t ω t d t t t For different shped profiles such s when t!! t t or! t, the shped input is different thn in (7) but the corresponding SOS control lws cn be derived similrly s detiled below for the shped time-optiml input in (7). Bsed on the x r nd stte trjectories due to the shped time-optiml input (7) t the switching times, t, t, nd t, we cn derive switching curves to be used in SOS control lw. Fig. 4 shows the phse-pe trjectories tht result from pplying the shped time-optiml input of (7) for severl positive mneuvers, nd it is cler tht it leds to five control regions. he first switching curve is when the shped input chnges its vlue from to t the switching time. he stte t this switching time is independent of the move distnce L. Hence, the switching curve is horizontl line (7)

4 elocity of Rigid Body (ẋr) S 4 3 x Error in Position of Rigid Body ( ) α S α S 3 α Fig. 4: Phse-pe trjectories (solid) nd switching curves (dshed) for move distnces in Cse. (see Fig. 4): S b e (8) Similrly, the lst switching curve is when the shped time-optiml input chnges its vlue from α to α t the switching time t. he stte t this switching time is independent of the move distnce L implying tht this switching curve is lso horizontl line (see Fig. 4) nd is α S 4 b e (9) he second switching curve occurs when the shped input switches from to α t the switching time t. he x r nd sttes t this switching time re both dependent on the move distnce L. It is necessry to isolte nd eliminte t nd L. First, solve for time t from the eqution t the switching time t : t b b e () S 4 We cn then solve for L from L b b e () L is the curve-fit function of the switching time t. sing () nd () in the expression for the x r stte t the switching time t, the second switching curve is eqution (), λ is the rel root of (). Similrly, we cn compute the third switching curve by solving for t nd L (see (3) nd (4)) from the expression for the stte t the switching time t, the shped input switches from α to α. hen substituting (3) nd (4) into the expression for the x r stte t the switching time t, we obtin the third switching curve s eqution (5), λ is the rel root of (4). When the x r nd sttes re below both the first nd lst switching curves, S nd S 4, it is necessry to determine which control, or α, to use. We seprte the first nd lst regions t the x r stte s described in (6) which is when the third switching curve S 3 intersects with the lst switching curve S 4. he derivtion of the switching curve equtions for mneuvers in the negtive direction cn be computed similrly. In fct, the switching curves for the fourth qudrnt re ntisymmetric to the switching curves for the second qudrnt (see Fig. 5). For rest-to-rest motion, the trjectory idelly does not go beyond the switching curve S 3 nd never enters the first or third qudrnt. However, s discussed in [6-9], in prctice, overshoot my occur due to implementtion issues or disturbnces. hus we ssign control vlues to these regions (see Fig. 5) so tht they will cuse the system to move into the second nd fourth qudrnt s quickly s possible. he gol is tht the control ction in the second nd fourth qudrnts brings the system to the origin in the respective qudrnt. S λ L S 3 λ t b b b b e b () b α α e b e (3) b α α e b e (4) b α α e α b b e (5) b x λ α α e e b e α b (6)

5 S S 3 f S S 4 x x f Fig. 5: Entire phse-pe: the control input is ssigned to be, f,, nd f in the first through fourth qudrnts, respectively. he SOS control lw for ll four qudrnts cn be expressed s the following logic expression: u sgn sgn sgn sgn sgn sgn f sgn sgn sgn sgn sgn sgn sgn sgn sgn sgn sgn f sgn sgn sgn sgn sgn sgn (7) f 4 A A CD C D α D C C D C C α D D α D D PD D A A CD C D α CD C D PB B A B A B α B B PB B (8) A sgn S B sgn x C sgn S D sgn S 3 E sgn S 4 P α nd x r E E x ref is the error in position of the rigid body. Switching curves S -S 4 nd x re given in (8), (9), (), (5), nd (6). f is the switching function for the second qudrnt, nd f is the switching function for the fourth qudrnt nd is f f (9).. Cse For Cse, the time-optiml control input (Fig. 3c) tht drives the second-order rigid body mode with dmping to the slew rte limit, holds it there, nd then decelertes to drive the rigid body mode to come to rest t the desired position is u t t b b t t 3 L b t t t b t t () α t t t 3 αb αb αb L b αb b b b b αb αb nlike Cse, the switching times t, t, nd t 3 cn be solved nlyticlly in closed-form rther thn using numericlly curve-fit solutions. Assuming tht the shper length of the input shper (4) is smller thn the time durtions t nd t 3 t of the first nd third control levels nd the move distnce L flls into the mneuver rnge such tht! t t, the shped time-optiml input for Cse is (see Fig. 3d) u t t t t t t t b α t b α t α t α t 3 t t t t t t 3 t t 3 () Similrly s done in Cse, we obtin the switching curves for Cse s shown in Fig. 6 nd the SOS control lw s in (7), f is now

6 elocity of Rigid Body (ẋr) S b S 3 S S 6 α 8 x Error in Position of Rigid Body ( ) α S 4 b α α Fig. 6: Phse-pe trjectories (solid) nd switching curves (dshed) for move distnces in Cse. f sgn sgn x B B α sgn x CG C G sgn B EH E H sgn G CD C sgn D sgn C 4 b αsgn sgn EG E G sgn C x Asgn A sgn D sgn A b α Esgn E Dsgn D sgn E sgn H A sgn S 6 B sgn S C sgn S D sgn S 5 E sgn G sgn S 3 H sgn S 4 S b S b e e S5 () S 3 S 4 b αb αb α b S 5 b S 6 x α b αb αb αb b b α αb αb e αb αb αb α b e e αb α b α b For rest-to-rest motion, the trjectory idelly does not go beyond the slew rte limit. However, s discussed in Cse, overshoot my occur in prctice due to implementtion issues. hus we ssign the SOS control lw () to yield the control vlue b α in the region bove the slew rte limit..3. Cse 3 he time-optiml control input for Cse 3 (Fig. 3e) is the sme s for Cse except tht the mneuver size L is now lrger. Shping the time-optiml input () using the shper s in (4) nd ssuming tht the shper length is smller thn the time durtions t nd t 3 t of the first nd third control levels of the time-optiml input nd the move distnce L flls into the mneuver rnge such tht & t t, we obtin the shped time-optiml input for Cse 3 s (see Fig. 3f) u t t t t t b t b b α t α t α t 3 t t t t t t ω t d t 3 t t 3 (3) Similrly s done in Cse, we obtin the switching curves for Cse 3 s shown in Fig. 7 nd the SOS control lw s in (7), f is

7 elocity of Rigid Body (ẋr) S b b b α α S α S 4 5 x 5 Error in Position of Rigid Body ( ) Fig. 7: Phse-pe trjectories (solid) nd switching curves (dshed) for move distnces in Cse 3. f sgn sgn x sgn x B B b E E sgn C C C sgn B b α E sgn E b E E sgn sgn x α Dsgn D b αasgn A sgn D α sgn A D D (4) S A sgn S 4 B sgn S C sgn S D sgn S 3 E sgn b S b α S 3 b S 4 x α b e αb e e αb αb α b e S3 ble : oice coil motor nd flexible structure prmeter vlues. he modl frequency is ω n f n rd/s. For these prmeter vlues, s nd b b 4 7 /( 3 s in the stte-spce model of (). Prmeters Disk Drive lues K t 9 8 N m/a K b 9 8 s K f 7 /( 3 s R 7 5 Ω J 5 6 kg m m mx min 7 7 f n 3 Hz ζ For the sme reson s in Cse, we ssign the SOS control lw (4) to yield the control vlue b α in the region bove the slew rte limit. 3. Simultion Results Figs. 8 nd 9 show typicl time responses of the SOS control pplied to disk drive red/write rm driven by voice coil motor which hs model s shown in Fig. with the system prmeter vlues given in ble the sttes x r x f nd ẋ f re mesured in rd, rd/s, 4 rd, nd rd/s, respectively. he shped voltge control commnd u sturtes t mx nd min, the decelertion fctor α 7. he slew rte limit is rd/s. he move distnces L re 8, 7, nd 8 rd which fll into the mneuver rnges of Cses -3, respectively. he simultions show tht the red/write rm is moved from rest to rest to the desired trck nd t the end of the move there is no residul vibrtion. he time-optiml controls for Cses nd 3 cn be determined nlyticlly. For Cse, however, we use polynomil curve-fit function to pproximte t since it cn not be solved nlyticlly from (6). With the curve-fit function for the switching time t, the remining equtions to determine the SOS control lw re determined nlyticlly s described in Section. he mximum errors of the residuls for the curve-fit polynomils of degree 3, 4, nd 5 re 9 3, 3, nd 8 9 4, respectively. he simultion in Fig. 8 uses the polynomil of degree 5. Note tht insted of hving one curve-fit function of t for the entire rnge of the move distnce in Cse, we cn improve the ccurcy of the curve-fit function by dividing the move distnce into severl rnges within Cse nd then computing the curve-fit functions nd the SOS control lws for ech rnge. his is more computtionlly complex nd should be done only when there re extreme ccurcy requirements.

8 u.8 u.7 xr 5 ẋr ẋr x f xr ẋ f 3 4 ime ( 3 s) Fig. 8: ypicl time response due to the SOS control for Cse. 4. Conclusions Shped time-optiml servomechnisms (SOS) hve been derived for flexible systems consisting of second-order rigid body with dmping nd dmped flexible mode nd hving different ccelertion nd decelertion cpbilities. Considering slew rte limit constrints when solving the timeoptiml control for the second-order rigid body with dmping leds to three rnges of mneuver sizes. While the closedform solution of the time-optiml control for the second-order rigid body with dmping portion of the first mneuver rnge is not known, the SOS control for this mneuver rnge cn still be nlyticlly derived by using the curve-fit pproximtion of the time-optiml control. Polynomil curve fits were found to give high ccurcy. References [] E. Brbieri nd. Ozguner. A New Minimum-ime Control Lw for One-Mode Model of Flexible Slewing Structure, IEEE rns. Aut. Ctrl., 38(), Jn [] W. J. Book. Controlled Motion in n Elstic World, ASME J. Dyn. Sys., Mes., nd Ctrl., 5(), June 993. [3] G. F. Frnklin, J. D. Powell, nd M. L. Workmn. Digitl Control of Dynmic Systems, Reding, MA: Addison-Wesley, 998. [4] H.. Ho. Fst Servo Bng-Bng Seek Control, IEEE rns. Mg., 33(6), Nov [5] J. L. Junkins nd Y. Kim. Introduction to Dynmics nd Control of Flexible Structures, AIAA, Wshington, D.C., 993. [6] C. L-orpchrpn nd L. Y. Po. Control of Flexible Structures with Projected Phse-Pe Approch, Proc. Amer. Ctrl. Conf., Arlington, A, June. [7] C. L-orpchrpn nd L. Y. Po. Shped Phse- Pe Control for Flexible Structures with Friction, Proc. Amer. Ctrl. Conf., Anchorge, AK, My. [8] C. L-orpchrpn nd L. Y. Po. Shped Control for Dmped Flexible Structures with Friction nd Slew Rte Limits, Proc. IEEE Conf. Decision nd Ctrl., Ls egs, N, Dec.. x f ẋ f u xr ẋr x f ẋ f ime ( 3 s) Fig. 9: ypicl time response due to the SOS control for Cse (top) nd Cse 3 (bottom). [9] C. L-orpchrpn nd L. Y. Po. Fst Seek Control for Flexible Disk Drive Systems with Bck EMF nd Inductnce, Proc. Amer. Ctrl. Conf., Denver, CO, June 3. [] F. Li nd P. M. Binum. Anlyticl ime-optiml Control Synthesis of Fourth-Order System nd Mneuvers of Flexible Structures, J. Guid., Ctrl., nd Dyn., 7(6), Nov.-Dec [] W. S. Newmn. Robust Ner ime-optiml Control, IEEE rns. Aut. Ctrl., 35(7), July 99. [] L. Y. Po nd G. F. Frnklin. Proximte ime-optiml Control of hird-order Servomechnisms, IEEE rns. Aut. Ctrl., 38(4), April 993. [3] L. Y. Po nd G. F. Frnklin. he Robustness of Proximte ime-optiml Controller, IEEE rns. Aut. Ctrl., 39(9), Sept [4] N. C. Singer nd W. P. Seering. Preshping Commnd Inputs to Reduce System ibrtion, ASME J. Dyn. Sys., Mes., nd Ctrl., (), Mr. 99. [5] M. L. Workmn nd G. F. Frnklin. Implementtion of Adptive Proximte ime-optiml Controllers, Proc. Amer. Ctrl. Conf., Att, GA, June 988.