Tuning nd Anlysis of Frctionl Order Controllers for Hrd Disk Drive Servo System Rkhi. S Dept. of Electricl nd Electronics Engineering Lourdes Mth College of Science nd Technology Thiruvnnthpurm, Indi Rohini G. P Dept. of Electricl nd Electronics Engineering Lourdes Mth College of Science nd Technology Thiruvnnthpurm, Indi Abstrct- Hrd-Disk-Drive Servo systems re mechtronic systems tht demnds for high precision control of output of the system. The hed positioning servo system is to mintin the red/write hed t desired trck (trck following). The loop gin vritions cused by different hrdwre components is mjor problem tht ffects the trcking control performnce of HDD s. To chieve consistent trcking performnce, the problem demnds for controller tht is robust to loop gin vritions. In this pper, frctionl order proportionl integrl (FO-PI) nd frctionl order proportionl integrl derivtive (FO-PID) controllers re proposed nd designed. Simultion results shows tht the proposed controllers cn ensure robustness ginst loop gin vritions nd the overll trcking control performnce is improved thn trditionl integer order controller. Key Words: Hrd- disk drive, frctionl order proportionl integrl (FO-PI) control, frctionl order proportionl integrl derivtive control (FO-PID), proportionl integrl derivtive controller (PID), robustness, isodmping I. INTRODUCTION Hrd Disk Drives (HDD s) re widely used dt storge medium in the modern er of digitl technology [1], [2]. The servo mechnism in HDD systems plys n importnt role in meeting the high performnce requirements of HDD s [3]. The servo mechnism includes Voice Coil Motor (VCM) ctutor which is used to position red/write hed ssembly model for recording nd retrieving dt. Trck following, trck seeking nd trck settling re the control modes of the servo mechnism. Trck seeking mens the heds to be positioned from one trck to desired trck. Trck Following needs the heds to be mintined s close s possible to desired trck nd follow it. In trck settling the heds re llowed to settle within the desired trck. Trck following requires tht the position error, i.e., the reltive difference between ctul hed position nd desired should be minimum. For high performnces, the HDD system should hve consistent trck following performnces under the influence of gin vritions tht cn be cused by externl disturbnces, chnges in externl tempertures [4]. Mny integer order controllers/compenstors [5], [6] hve been designed in order to solve the problem. Even though some of the requirements re stisfied, these controllers re not found robust ginst loop gin vritions. The systems designed with integer order controllers my exhibit vritions in time domin nd frequency domin requirements with gin vritions. An isodmping property is not exhibited with designed integer order controller. Such performnce cnnot be ccepted in mechtronic system like HDD s were high precision control re necessry. Severl control pproches such s Liner Qudrtic Gussin control, robust control, H control hve been proposed for the trck following performnces. [7-9]. In [10], [11] the effects of loop gin vritions re suppressed using utomtic gin control methods. More control methods should be there tht should come up with high ccurcy nd simplicity in control. In the men while frctionl order controllers (FO controllers) hve got wide spred ttrction in control engineering nd technology [12], [13]. In the pst the FO controllers were not used becuse of unfmilirity with the use of FO prmeters, nd the bsence of necessry computtionl power. But the frctionl order differentil equtions could well define the dynmic processes nd so the computtionl progresses cn be mde esier. With the introduction of frctionl order prmeters more ccurcy in control cn be obtined [14]. The min contributions of this brief include: Frctionl order proportionl integrl nd frctionl order proportionl integrl derivtive controllers re proposed nd designed for the HDD- VCM ctutor model. Certin tuning constrints re imposed to ensure tht the gin cross over nd phse mrgin specifictions re met nd robustness ginst gin vritions re chieved. The FO controller performnces re compred with n integer order PID controller. All the controllers re ddressed under sme design specifictions. From the simultions it is cler tht the FO controllers outperform the PID controllers. This brief provides n illustrtion of benefits of pplying FO-PI nd FO-PID controllers in the rel HDD industry. The outline of rest of this brief is orgnized s follows. Mthemticl modelling of VCM ctutor is described in section II. Section III explins the bsic concepts of frctionl clculus. In section IV initil design procedures for the FO controllers re presented. Some tuning constrints re imposed in order to ensure tht the gin cross over frequency, phse mrgin nd robust specifictions re met. Then FO-PI controller design procedures re given. For the FO-PID controllers, new tuning rule bsed on Ziegler- Nichols nd 239
Astro m- H gglund methods re proposed. In section V the implementtion detils of the FO controllers on the HDD ctutor model re given. In section VI the simultion results tht re crried out to compre nd evlute the trcking performnces of FO controllers with integer order PID controllers re presented. Conclusions re given in section VII. II. HDD ACTUATOR MODELLING In this section we present the dynmic model of HDD with the Voice Coil Motor (VCM) ctutor [1]. The fig. 1 shows the circuit model of VCM ctutor. TABLE I PLANT MODEL PARAMETERS Prmeter Description Vlue J Moment of 0.2 Inerti k t Torque constnt 20 k y Position 10000 mesurement gin ω n Resonnt 80 Frequency Coupling 0.0032 coefficient b Coupling coefficient 0.25 where, θ s is the hed position, which is given in rdins. It cn be trnsformed into dt trcks nd the modified trnsfer function is: The ctutor is ttched to the coil of VCM nd is pivoted. When current is pssed through the coil of motor, n electromgnetic force is experienced. This cuses torque on the rotry ctutor. The torque of the ctutor is given by: τ = k t I c where, I c is the current through the coil. The eqution of motion of rotry ctutor is: τ = Jθ where, J is the moment of inerti of moving prts, nd θ being the ngulr displcements. is Fig.1 Voice Coil Motor (VCM) Actutor From (1) nd (2) the mechnicl dynmics of the ctutor τ = Jθ k t For the movement of the ctutor, it needs to be powered by n mplifier. Current mplifiers re commonly used due to its high impednce. The simplified trnsfer function of mplifier is given by: I s u s = k sτ +1 where, τ is the torque constnt n k is the mplifier gin. I(s) nd u(s) re the current input nd outputs to the mplifier. Thus the trnsfer function of VCM with current driver is: θ s = k t k 1 u s J s 2 (1) (2) (3) (4) (5) y s u s = k k y s 2 Since the ctutor structures re not perfectly rigid nd contin severl flexible modes tht cn cuse resonnces nd vibrtions. So the effects of such modes cnnot be neglected. More ccurte model is obtined by considering the resonnce effects. The overll ccurte model is given s: P s = k k y s 2 ω n s+bω 2 n s 2 +2ζω n s+ω 2 n where, k is the ccelertion constnt, k y is the position mesurement gin. ω n is the resonnt frequency. nd b re the resonnt coupling coefficients. The prmeters for the plnt model re given in Tble I. Then the model described in (7) cn be written s P s = 1.287e8s+8.085e8 s 4 +502.7s 3 +2.527e5s 2 III. BASIC CONCEPTS OF FRACTIONAL CALCULUS The frctionl order clculus is n re tht dels with derivtives nd integrls from non integer orders. The bsic continuous integro-differentil opertor is defined s [15-18] D t α = d α dt α, R α > 0 1, R α = 0 t dτ α, R α < 0 where α is the order of the opertion; generlly α R nd is constnt corresponds to initil conditions. (6) (7) (8) (9) 240
The most common Cputo definition for differentition is α D t f(t) = 1 Γ n α t where n is n integer nd n-1< α<n f n τ t τ α +1 n Similrly Cputo s frctionl order integrtion is given s D t α+1 n f(t) = 1 Γ n α 1 t dτ f τ t τ α +1 n +1 IV. FRACTIONAL ORDER CONTROLLER DESIGN In this section the generl form of FO controllers nd the design strtegies re presenting. Generlly the FO controllers re modeled s: C 1 s = K p 1 + K i s λ dτ (10) (11) (12) 5. Obtin λ nd K i from the intersection point between Curves 1 nd 2. 6. Clculte K p from form specifiction 1 By these tuning rules nd specifictions, FO-PI controller prmeters cn be obtined. B. Frctionl order PID controller In this section the tuning strtegies for n FO-PID controller is presented. For the FO-PID controller of the form in (13) there re five prmeters to obtin. A tuning rule bsed on Ziegler- Nichols [19] nd Astro m- H gglund [20] method is dopted here. The rules re summrized s follows: 1. Obtin K p nd K i from Ziegler- Nichols tuning rule. 2. Obtin K d from Astro m- H gglund Method. 3. Solve for λ nd µ using (14) - (16) C 2 s = K p + K i s λ + K d s μ (13) V. FRACTIONAL ORDER CONTROLLER IMPLEMENTATION ON HDD MODEL where, (12) nd (13) represents the generl form of n FO-PI nd FO-PID controller. For the plnt model described in eqution (8) the FO-PI /FO-PID controllers cn be designed s per the three design specifictions given below. For the controller of the form The design specifictions to met re: 1) Gin cross over frequency specifictions. C jω gc P jω gc = 1. 2) Phse mrgin specifictions: (14) A. Implementtion of FO-PI controller The mgnitude nd phse of the plnt model is given s: P jω = < P jω = π tn 1 3199.999 1+ 0.159154943 ω 2 ω 2 1 3.95785836 e 6ω 2 2 + 1.9894368 e 3ω 2 1.989436789 e 3ω 1 3.957858736 e 6ω 2 (17) + tn 1 0.159154943ω (18) For the FO-PI controller of the form (12) the mgnitude nd phse re: < (C jω gc P jω gc = φ m π (15) C 1 jω = 2 + Ki ω λ sin λπ 2 2 where φ m is the desire phse mrgin nd ω gc is the gin cross over frequency. K p 1 + K i ω λ (cos λπ 2 (19) 3) Robustness to gin vritions < C 1 jω = tn 1 K iω λ sin λπ 2 1+K i ω λ cos λπ 2 (20) d< C jω cg P jω cg dω = 0 (16) Then from specifiction 2 i.e., the derivtive of the phse of the open-loop system with respect to the frequency is forced to be zero t the gin crossover frequency so tht the closed-loop system is robust to gin vritions, nd therefore the time responses of the systems re lmost invrint. A. Frctionl order PI controller The tuning strtegies for the FO-PI controllers re s follows: 1. Given ω gc, the gin crossover frequency 2. Given ϕ m, the desired phse mrgin. 3. Plot curve 1 corresponding to K i with respect to λ, ccording to specifiction 2 4. Plot curve 2 corresponding to K i with respect to λ, ccording to specifiction 3 K i ω gc λ sin λπ 2 1+K i ω gc λ cos λπ 2 = tn < P jω gc φ m + π From 21, we obtin, where, K i = c bc c = tn < P jω gc φ m + π = ω gc λ sin λπ 2 (21) (22) (23) (24) 241
B. Implementtion of FO-PID controller For the FO-PID controller of the form in (13).The mgnitude nd phse of the controller re given by (33) nd (34). C 2 jω gc = A 1 2 + B 1 2 (33) < C 2 jω gc = tn 1 B 1 A 1 (34) where, A 1 = K p + K i ω gc λ cos λπ 2 + K dω gc μ cos μπ 2 (35) From specifiction 3 d dω gc Fig. 2 K i V/s λ plot b = ω gc λ cos λπ 2 tn 1 K iω λ gc sin λπ 2 1+K i ω λ gc cos λπ 2 From this K i cn be evluted s K i = df e ± df e 2 4gd 2 2dg = d < P jω dω gc gc (25) (26) (27) B 1 = K i ω gc λ sin λπ 2 + K dω gc μ sin μπ 2 From specifiction (1) From specifiction (2) From specifiction 3 A 1 2 + B 1 2. P(jω gc = 1 tn 1 B 1 A 1 +< P jω gc = φ m π d tn 1 B 1 = d < P jω dω gc A 1 dω gc gc (36) (37) (38) (39) where, d = d dω gc < P jω gc e = λω λ 1 gc sin λπ 2 f = 2ω λ gc cos λπ 2 ; g = ω gc λ 2 (28) (29) (30) The vlues of K p nd K i obtined from Ziegler- Nichols tuning rule re 0.58477 nd 94.133. Then from Astro m- H gglund method K d is obtined s 0.0019. The vlues of λ nd µ re solved using fsolve commnd in MATLAB nd re obtined s 1.2913 nd 1.3556 respectively. Then the FO-PID controller hs the form: C 2 s = 0.58477 + 94.13s 1.2913 + 0.01896s 1.3556 Then vlue of K p is obtined from: K p = 1 P jω gc 1+K i ω gc λ (cos λπ 2 2 + sin λπ 2 2 The curves between λ nd K i re shown in Fig.2. From the intersections, the optimum vlues of λ nd K i re obtined s 0.2479 nd 1.057. Then, the designed FO-PI controller is of the form C 1 s = 0.085 1 + 1.057s 0.2479 (31) (32) VI. SIMULATION RESULTS (40) This section presents the vlidtion of the vrious performnces with FO controllers. We hve lso designed n integer order controller for fir comprison with FO controllers. The performnces with ech controller under gin vritions re evluted. Fig. 5 shows trck following performnces of the FO controllers nd n integer order PID controller under gin vritions. The response chrcteristics cn be nlyzed from Tbles II, III nd IV 242
TABLE II TRACKING PERFORMANCE CHARACTERISTICS WITH PID CONTROLLER PID 90% gin 100% gin 110% gin Overshoot% 16.2467 15.1036 13.3772 Settling time(seconds) 0.1823 0.2158 0.2705 TABLE III TRACKING PERFORMANCE CHARACTERISTICS WITH FOPI CONTROLLER () FOPI 90% gin 100% gin 110% gin Overshoot% 8.7003 8.4464 8.3916 Settling time(seconds) 0.2587 0.2597 0.2609 TABLE IV TRACKING PERFORMANCE CHARACTERISTICS FOPID CONTROLLER FOPID 90% gin 100% gin 110% gin Overshoot% 6.8495 6.5760 6.0150 Settling time(seconds) 0.1355 0.1332 0.1303 (c) Fig 5. Trck following performnces under gin vritions () PID controller FO-PI controller (c) FO-PID controller From the tbles we cn infer tht the settling time nd overshoots re lmost similr under gin vritions for FO controllers. Isodmping property is exhibited with these controllers. While for the integer order PID controller the trcking performnces re not so robust under gin vritions. With the FO controllers overshoots re reduced to 44%-56 %.Among the FO controllers FO-PID controllers shows better trck following performnces when compred to FOPI controller. With the FO-PID controllers there is 38% reduction in settling time. Now the Open loop Bode plot with gin vritions of ech controller is shown in fig.6. The Trck following error plot with ech controller re shown in fig.7. From figure 6 we cn infer tht the FO controllers mintin the flt phse feture which shows tht they re more robust to gin vritions. The phse mrgin nd gin cross over frequencies re lmost similr with gin vritions. While with PID controllers, these my vry.fig.7 shows the trck following error with ech of the controllers. The error is regulted in minimum time with FO controllers while it is 0.7736 in the cse of PID controllers. 243
() () (c) (c) Fig. 6 Open loop Bode plot comprison with gin vritions () PID controller FO-PI controller (c) FO-PID controller Fig 7. Trck Following error () PID controller FO-PI controller (c) FO-PID controller 244
VII. CONCLUSION In this brief two frctionl order controllers (FO-PI nd FO-PID) controllers re proposed nd designed for n HDD model so s to chieve the gin crossover frequency, phse mrgin nd robustness requirements. FO controller implementtion detils on the HDD model were clerly presented. The trck following performnce nd open loop Bode response under gin vritions re evluted with the FO controllers. The performnces re compred with n integer order PID controller. From the simultion results it is cler tht the FO controllers re more robust under gin vritions. The trck following performnces is consistent with FO controllers. REFERENCES [1] A.A. Mmun, G. Guo, nd C. Bi, Hrd Disk Drive: Mechtronics nd Control, Boc Rton, FL, USA: CRC Press, 2007. [2] K. Peng, B. M. Chen, T. H. Lee, nd V. Venktrmnn, Hrd Disk Drive Servo Systems (Advnces in Industril Control), 2nd ed. New York, USA: Springer-Verlg, 2006. [3] C. K. Thum, C. Du, B. M. Chen, E. H. Ong, nd K. P. Tn, A unified control scheme for trck seeking nd following of hrd disk drive servo system, IEEE Trns. Control Syst. Technol., vol. 18, no. 4, pp. 294 306, Mr. 2010 [4] F. C. Wng, Q. W. Ji, C. F. Wng, nd J. Y. Wng, Servo loop gin clibrtion using model reference dpttion in HDD servo systems, in Proc. Chin. Control Decision Conf., Guilin, Chin, Jun. 2009, pp. 3377 3381. [5] G. F. Frnklin, J. D. Powell, nd M. L. Workmn, Digitl Control of Dynmic Systems, 3rd ed. Reding, MA, USA: Addison-Wesley, 1998. [6] H. Fujimoto, Y. Hori, T. Yrnguchi, nd S. Nkgw, Proposl of seeking control of hrd disk drives bsed on perfect trcking control using multirte feedforwrd control, in Proc. 6th Int. Workshop Advnced Motion Contr., Ngoy, Jpn, 2000, pp. 74 79. [7] H. Hnselmnn nd A. Engelke, LQG-control of highly resonnt disk drive hed positioning ctutor, IEEE Trns. Ind. Electron.,vol. 35, pp. 100 104, Jn. 1988. [8] T. B. Goh, Z. Li, B. M. Chen, T. H. Lee, nd T. Hung, Design nd implementtion of hrd disk drive servo system using robust nd perfect trcking pproch, IEEE Trns. Contr. Syst. Technol., vol. 9, pp. 221 233, Mr. 2001. [9] M. Hirt, K. Z. Liu, T. Mit, nd T. Ymguchi, Hed positioning control of hrd disk drive using H theory, in Proc. 31st IEEE Conf. Decision Contr., Tucson, AZ, 1992, pp. 2460 2461. [10] E. Bnt, Anlysis of n utomtic gin control (AGC), IEEE Trns.Autom. Control, vol. 9, no. 2, pp. 181 182, Apr. 1964. [11] Q. W. Ji nd G. Mthew, A novel AGC scheme for DFE red chnnels, IEEE Trns. Mgn., vol. 36, no. 5, pp. 2210 2212, Sep. 2000. [12] Y. Q. Chen, I. Petr s, nd D. Y. Xue, Frctionl order control A tutoril, in Proc. Amer. Control Conf., St. Louis, MO, USA, 2009, pp. 1397 1411. [13] I. Petr s, The frctionl-order controllers: Methods for their synthesis nd ppliction, J. Electr. Eng., vol. 50, nos. 9 10, pp. 284 288, 1999. [14] I. Petráš, Stbility of frctionl-order systems with rtionl orders: A survey Frctionl Clculus & Applied Anlysis, vol. 12, no. 3, pp. 269-298, 2009. [15] K. B. Oldhm, nd J. Spnier, Frctionl Clculus: Theory nd Applictions of Differentition nd Integrtion to Arbitrry Order. NewYork London: Acdemic Press, 1974. [16] K. S. Miller, nd B. Ross, An Introduction to the Frctionl Clculus nd Frctionl Differentil Equtions. New York, USA: John Wiley nd Sons, 1993. [17] I. Podlubný, Frctionl Differentil Equtions. Sn Diego, CA, USA: Acdemic Press, 1999. [18] R. E. Gutiérrez, J. M. Rosário, nd J. T. Mchdo, Frctionl Order Clculus: Bsic Concepts nd Engineering Applictions Mthemticl Problems in Engineering, vol. 2010, 19 p., 2010, DOI:10.1155/2010/375858. [19] K. Ogt,: Modern control engineering (Prentice-Hll, New Jersey,2002) [20] K.J. Astro m nd T.H gglund, PID controllers: theory, design nd tuning (Instrument Society of Americ, North Crolin, 1995) 245