Sliding Mode and PID Control of a Dual Stage Actuator for Precision Positioning

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1 Preprints of the 19th World Congress The International Federation of Automati Control Cape Town, South Afria. August 24-29, 214 Sliding Mode and PID Control of a Dual Stage Atuator for Preision Positioning Shingo Ito Juergen Steininger Georg Shitter Automation and Control Institute (ACIN), Vienna University of Tehnology, Gusshausstrasse 27-29, 14, Vienna, Austria ( {ito, juergen.steininger, shitter}@ain.tuwien.a.at) Abstrat: Atuators omposing dual stage atuation usually have different purposes and mehanisms to omplement eah other. Hene, they potentially have different types of ontrol hallenges to ahieve high-preision motion performane. This paper presents the use of singleinput single-output ontrollers to effiiently overome the potential problems by seleting proper ontrol tehniques to eah atuator. The setup is a low-stiffness dual stage atuator apable of positioning with nanometer resolution without additional vibration isolation. The system onsists of a CD/DVD laser pikup and a linear motor with roller bearings, whih suffer from environmental vibrations and frition, respetively. By seleting and implementing a tamed PID ontroller for the pikup and a sliding mode ontroller for the linear motor, the dual stage atuator is able to move over 1 mm with a maximum veloity of.34 m/s and position with a stati preision of ± 2.5 nm (peak-to-peak). 1. INTRODUCTION In appliations where high-preision positioning over a long range is required, a single atuator may not be able to ahieve the desired performane. To satisfy the requirements, a long-stroke oarse atuator an be ombined with a short-stroke fine (high-preision) atuator as a dual stage atuator (DSA). This tehnique has been suessfully applied to many appliations, suh as optial disk drives (Chaghajerdi [28]) and wafer sanners for lithography (Butler [211]). In order to ontrol DSA systems, multi-input multi-output (MIMO) ontrol design an be applied, sine they an be regarded as a system with at least multiple inputs. The MIMO ontrol inludes H 2 ontrol, H ontrol, µ-synthesis, and sliding mode ontrol (SMC) (Hu et al. [1998], Al Mamun et al. [23], Tai and Chen [25], Nagamune et al. [26]). These ontrols an guarantee the stability and robustness of the overall system. Another strategy to ontrol DSA systems is to design single-input single-output (SISO) ontrollers for the fine and oarse atuators in a ertain onfiguration (Guo et al. [1999]). By using the SISO ontrol design, the order of ontrollers an be lower than that of MIMO (Semba et al. [1999]). More importantly, the SISO design has the freedom of seleting the ontrol design tehniques individually for the fine atuator and the oarse atuator, whereas the MIMO usually applies same tehniques to both. In fat, Herrmann et al. [27] suessfully applied disrete SMC to the fine atuator, while using proximate time-optimal ontrol for the oarse atuator in a hard disk drive (HDD). In general, the fine atuator and the oarse atuator of a DSA have different atuation priniples and mehanisms to omplement eah other. For example, a piezoeletri stage is mounted onto a gantry stage with ball srews and DC servo motors to ompensate for baklash and frition (Lin et al. [29]). A piezoeletri devie is also installed as the fine atuator on a voie oil motor to inrease the ontrol bandwidth in a HDD. In a wafer sanner, short-stroke Lorentz atuators are used together with linear motors, physially isolating disturbanes suh as floor vibrations (Butler [211]). Due to suh multiple atuation priniples potentially existing in a DSA system, it is natural that different types of ontrol hallenges arise individually for the fine atuator and the oarse atuator. To overome suh problems for high-preision positioning over a long range, the SISO ontrol design an be effetively applied by utilizing the freedom of ontrol tehnique seletion. In other words, the SISO design allows the user to pik proper tools to individually solve the distint ontrol problems. This paper presents a low-stiffness DSA, whih is onstruted to ahieve long-range positioning with nanometer resolution without extra vibration isolation (e.g. an optial table) (Ito et al. [213]). This system onsists of a laser pikup of a CD/DVD player as the fine atuator and a linear motor with roller bearings as the oarse atuator. For these atuators, ontrollers are individually designed to ahieve high-speed high-preision positioning over a long range. The rest of this paper is organized as follows. Setion 2 introdues the DSA system. This system is modeled and its disturbanes are analyzed in Setion 3. Controllers are designed in Setion 4. Setion 5 presents experimental results, omparing performane of the SISO ontrol design with and without the individual ontrol tehnique seletion. The paper is onluded in Setion SYSTEM DESCRIPTION Fig. 1 shows a DSA setup built on a pik-and-plae mahine (CAT(2a), Philips, Amsterdam, Netherlands), with- Copyright 214 IFAC 655

2 19th IFAC World Congress Cape Town, South Afria. August 24-29, 214 xf Linear motor Fine atuator Coarse atuator k m mf Ff x F - Ff Pikup and retrorefletors x Detetors of interferometer Fig. 2. Lumped mass model of DSA. Fig. 1. Photograph of the DSA setup. out external vibration isolation suh as an optial table. This mahine has six linear motors, and one of them is used as the long-stroke oarse atuator of the DSA. The linear motor is guided by roller bearings and operated by a servo driver in fore ontrol mode. As the fine atuator, a Lorentz atuator in a laser pikup (SF-HD65, Sanyo, Osaka, Japan) is used and driven by a voltage amplifier. The atuation range of the fine atuator is limited to approximately ± 1 mm. To optially measure the fine atuator position, its objetive lens is replaed by a ube-orner retrorefletor, and an interferometer (1899A, Agilent Tehnologies, San Franiso, USA) is mounted on the platform for the real time ontrol of the DSA. The interferometer has a resolution of 1.25 nm and detets movement up to.4 m/s. An additional retrorefletor is mounted on the oarse atuator to measure its position with a seond interferometri detetor. This seond sensor is not used for the ontrol, but for evaluation only. The servo driver, the voltage amplifier and the interferometers are all onneted to a rapid ontrol prototyping system (DS15(CPU), DS523(FPGA), DS212(DAC), dspae GmbH, Paderborn, Germany), where ontrollers are implemented running at a sampling frequeny of 2 khz. This system is also used for data aquisition. 3. SYSTEM ANALYSIS 3.1 Dynami model Fig. 2 shows a lumped mass model of the DSA setup, where xf and x are the positions of the fine and oarse atuators, respetively. Their fores are expressed as Ff and F, and k and are the stiffness and damping of the fine atuator. The moving mass of the fine atuator with the retrorefletor is mf and is about.7 g. The total mass of the oarse atuator s moving part inluding the stator of the fine atuator is m, and it is approximately 8.5 kg. To relate the atuator fores with the atuator drivers, two onstants Kaf and Ka are introdued as Ff = Kaf uf, F = Ka u, (1) where uf and u are the referene inputs of the drivers for the fine and oarse atuation, respetively. Using these inputs, the atuator positions an be expressed in the following equations under an assumption that m is suffiiently larger than mf (Ito et al. [213]): Xf (s) = Pf (s)(kaf Uf (s) Pd (s)x (s)), (2) X (s) = P (s)ka U (s), (3) where Xf (s), X (s), Uf (s) and U (s) are the Laplae transformations of xf, x, uf and u, respetively. Pf (s) and P (s) are transfer funtions that dominate the fine and oarse atuation, and Pd (s) represents the mehanial oupling of the two atuators. These transfer funtions are given as Pf (s) = Xf (s)/ff (s) = (mf s2 s k) 1, (4) P (s) = X (s)/f (s) = (m s2 ) 1, Pd (s) = s k. (5) (6) 3.2 Parameter estimation The unknown parameters in Setion 3.1 are estimated by measuring orresponding frequeny response data and fitting a model. Fig. 3 shows a Bode plot of the fine atuator and the fitting result using the fine atuator model Kaf Pf (s). The model shows a good fit at least up to 1 khz. It does not apture the seond mehanial resonane at a relatively high frequeny (around 6 khz) to keep its order low. Additional phase lag at high frequenies (> 1 khz) is due to a pole reated by the resistane and indutane of the pikup s oil. Beause the urrent through the oil is proportional to its voltage up to these high frequenies, a voltage amplifier is used for the fine atuator as mentioned. A Bode plot of the oarse atuator and urve fitting based on the ourse atuator model Ka P (s) are shown in Fig. 4, where the dynami system is well aptured approximately up to 1 Hz. A mehanial resonane at about 24 Hz is not modeled to keep the model simple for ontrol design. The estimated parameters are listed in Table 1. Table 1. Estimated parameters Parameter k Kaf Ka Value Unit N/m N/(m/s) N/V N/V 3.3 Disturbane of fine atuation To ahieve high-preision positioning with nanometer resolution at stati positions, the fine atuator needs to rejet environmental vibrations. For the evaluation of the disturbanes to be rejeted by the fine atuator, its position xf is measured while both atuators are turned off. The measured positional data in the time domain and the frequeny 6551

3 Cape Town, South Afria. August 24-29, 214 Mag. [db] (m/v) Phase [deg] Measured Fit Frequeny [Hz] Fig. 3. Bode plot of fine atuator (blue solid line) and urve fitting result (red dashed line). Mag. [db] (m/v) Phase [deg] -5-1 Measured Fit Frequeny [Hz] Fig. 4. Bode plot of oarse atuator (blue solid line) and urve fitting result (red dashed line). domain are shown in Fig. 5. The spetrum analysis shows that the major frequeny omponents are below 8 Hz. 3.4 Disturbane of oarse atuation For high-preision positioning, the fine atuator follows the position referene. Meanwhile, the oarse atuator arries the stator of the fine atuator, suh that the oil of the fine atuator always stays within its uniform magneti field. By doing so, the fine atuator an be operated in its linear range over the full DSA stroke. Therefore, to ahieve high-speed positioning, the oarse atuator needs to move fast enough, while its auray an be as poor as submillimeters for the fine atuator stroke (± 1 mm). For this reason, the major disturbane of the oarse atuator is frition due to the roller bearings, rather than the environmental vibrations. Frition an be desribed as a ombination of the stati, Coulomb and visous model (Armstrong-Hélouvry et al. [1994]). To evaluate the roller bearings, it is presumed that its visous frition is not a major disturbane soure, as damping effets annot be obviously seen in Fig. 4. Position x f [nm] Amplitude [nm rms] Time [s] Frequeny [Hz] Fig. 5. Fine atuator position x f (top) and its power spetrum (bottom) while both atuators are disabled. Thus, the maximum frition is measured as the fore required to start rolling the bearing balls. This required fore is externally applied and measured by a sensor (FSG- 15N1A, Honeywell, New Jersey, USA). The measurement is onduted 5 times at different positions. The results show that the frition depends on the oarse atuator position and varies between 8.2 N and 25.9 N. The mean value of the 5 measurements is 18.5 N. The measured maximum frition is onsidered at the ontroller design in Setion Observer 4. CONTROLLER DESIGN As mentioned in Setion 2, only the fine atuator position x f is measured for the real time ontrol. To regulate the oarse atuator, its veloity and position are estimated by a Kalman filter. The Kalman filter is designed based on a state-spae model orresponding to (2) and (3), that is ẋ ss = Ax ss Bu ss, y = Cx ss, (7) using x ss = [ x f x ẋ f ẋ ] T, u ss = [ u f u ] T, 1 1 A = k/m f k/m f /m f /m, f B = K af /m f K a /m, C = [ 1 ], (8) where y is the output. Note that this model is observable if k. In the ase of a zero-stiffness DSA (i.e. k = ), an additional sensor is required for the DSA operation. 4.2 Feedbak ontrol of fine atuator Sine the fine atuator has the first mehanial resonane at a low frequeny and shows a phase of 18 up to high frequenies in Fig. 3, its feedbak ontroller should provide 6552

4 Cape Town, South Afria. August 24-29, 214 Magnitude [db] 5-5 Measured Simulated Frequeny [Hz] Fig. 6. Measured (blue solid line) and simulated (red dashed line) sensitivity funtion of fine atuation. a phase lead to ahieve both high ontrol bandwidth and good disturbane rejetion. Furthermore, it is preferred that the ontroller is low order for the simpliity of implementation. Therefore, a PID ontroller with a low pass filter (tamed PID) is seleted for the fine atuation (Munnig Shmidt et al. [211]). This ontroller is tuned based on the fine atuator model K af P f (s), suh that its open-loop rossover frequeny is 1 khz with a suffiient phase margin. Fig. 6 shows the sensitivity funtion of the losed-loop fine atuation. Saturation an be a problem with short-stroke atuators, where anti-windup ontrol an improve stability (Kong et al. [21]). For preision positioning, however, the saturation has to be avoided by keeping the fine atuator within its linear operation range, whih requires an aording ontrol of the oarse atuator (f. Fig. 9). 4.3 Feedbak ontrol of oarse atuator For the oarse atuation, SMC is used as the feedbak ontroller to ompensate for the frition. SMC is nonlinear ontrol and has an exellent performane on disturbane rejetion. Partiularly when the mathing ondition is satisfied, the losed-loop system an be invariant to the disturbanes in ideal ases. Moreover, the ontroller design requires only the upper bound of the disturbanes. In this setion, the SMC design is divided into three steps: augmentation of the oarse atuator model, swithing surfae design based on the behavior of the system in the sliding mode, and ontrol input design to enfore the sliding mode (Shtessel et al. [213]). The design of SMC is disussed based on Fig. 7, whih shows the oarse atuator model and the SMC struture for the design. First of all, the oarse atuator model P (s) with K a is desribed as a state-spae model with the frition fore as disturbane d : ẋ ss = A x ss B u u B d d, (9) using x ss = [ x ẋ ] T, [ ] [ ] [ ] 1 A =, B u = K a m 1, B d = m 1, (1) For the SMC design, this state-spae model is augmented with an integrator and a low pass filter (LPF). The integrator integrates the traking error e of the oarse atuation. By inorporating this integrator, the oarse atuator is able to trak its referene r without a steadystate error. The LPF is added as a prefilter of the oarse atuator to mitigate the hattering phenomenon by attenuating unmodeled mehanial resonanes (Ito and Nonami [1997]). Sine the first unmodeled mehanial resonane is around 24 Hz in Fig 4, its utoff frequeny should be suffiiently lower than the resonant frequeny. Thus, the LPF is designed, suh that its utoff frequeny is 7 Hz, and is implemented in the form of a seond-order statespae model as shown in Fig. 7. For the SMC design, the augmented model is expressed in the following state equation: ẋ a ss = A a x a ss B a u u a B a r r B a d d, (11) where x a ss is a 5 1 state vetor. As the seond step of the design, a linear swithing surfae is introdued as the following manifold: σ(x a ss ) = Sx a ss =, (12) where S is a 1 5 matrix. While the system is fored to the sliding mode, σ = and σ = are satisfied. Using the latter ondition and (11) with d =, the equivalent ontrol u eq an be obtained as u eq = (SB a u ) 1 (SA a x a ss SB a r r ). (13) Substitution of (13) for the ontrol input of (11) with d = yields the system motion restrited to the swithing surfae: ẋ a ss = (I B a u (SB a u ) 1 S)(A a x a ss B a r r ). (14) By seleting x as the output of (14), the ontrol bandwidth of the system during the sliding mode an be obtained from the frequeny response from r to x. Matrix S is determined, suh that the ontrol bandwidth is around 15 Hz, whih orresponds to the bandwidth ahieved by a lead-lag ompensator for benhmarking (Setion 4.5). As the third step of the design, the ontrol input u a is seleted as a ombination of the equivalent ontrol (13) and a swithing ontrol u sw : u a = u eq u sw. (15) u sw enfores the sliding mode to the system and is omposed of a relay with a gain and a linear ontinuous feedbak as it follows: u sw (σ(x a ss )) = k sw1 sgn(σ(x a ss )) k sw2 σ(x a ss ), (16) where k sw1 and k sw2 are onstants. To determine their values, a Lyapunov funtion V = σ 2 /2 is onsidered. For the existene and the stability of the sliding mode, the following equation needs to be satisfied: V = σ(x a ss ) σ(x a ss ) = σ(x a ss )Sẋ a ss <. (17) Using (11)(15)(16), (17) leads to the following satisfatory ondition: k sw1 σ(x a ss ) k sw2 > (SB a u ) 1 SB a d d max, (18) where d max is the maximum frition measured in Setion 3.4. Finally, the relay funtion in (16) is smoothed by a sigmoid funtion, whih prevents the hattering by restriting the system not to the swithing surfae (12), but to its neighborhood. The sigmoid parameter, k sw1 and k sw2 are fine-tuned at the ontroller implementation, satisfying (18). 4.4 Overall ontrol of DSA Fig. 8 shows overall ontrol bloks of the DSA system. The DSA is deoupled by a prefilter K 1 af P d(s) 1, and the designed SISO ontrollers are applied to the orresponding atuators. Observer, PID and SMC bloks are the 6553

5 Cape Town, South Afria. August 24-29, 214 r d x lpf = A lpf x lpf B lpf u lpf y lpf = C lpf x lpf Low pass filter u a Augmented model u sw u eq y lpf σ K a - e ſdt x P du/dt x Coarse atuator Control law x a_ss Fig. 7. Coarse atuator model and SMC struture for ontrol design. The olored bloks are implemented in the SMC blok of Fig. 8. r G ff 1-α x f K af -1 P f -1 α - PID K - u af f -1-1 K Pd af P d x u r SMC K a P x x _ss Observer x lpf DSA (plant) SISO ontroller Fig. 8. Overall ontrol blok diagram of DSA. Kalman filter, the tamed PID and the SMC presented in the previous setions, respetively. To ahieve highspeed positioning to trak the referene r, K 1 af P f (s) 1 and G ff (s) are added for feedforward ontrol. G ff (s) is an input-shaping filer and is the inverse of the transfer funtion obtained from (14) by seleting x as the output. To generate the oarse atuation referene r, a parameter α tunable between and 1 is introdued. With α =, the oarse atuator traks r, while it follows the fine atuator (i.e. master-slave mode) by setting α = 1. For experiments, α is set at.5 to a first approximation. 4.5 Controller for benhmarking Benhmark ontrollers are designed for omparison. Although they use the same tamed PID for the fine atuator, a lead-lag ompensator is used for the oarse atuation (f. Ito et al. [213]). The ompensator is designed based on the oarse atuator model K a P (s), and is fine-tuned to maximize its ontrol bandwidth at the implementation, whih is about 15 Hz. The DSA onfiguration is slightly different in that the master-slave ontrol is used (α=1) with feedforward to the oarse atuator. Note that both feedbak ontrollers are seond-order filters in this benhmark system. 5. EXPERIMENTAL RESULTS Positioning using a minimum jerk trajetory (Yamaguhi et al. [211]) of 1 mm is onduted twie with the DSA system. The first experiment is relatively slow positioning P f x f Referene r [mm] Error (r - x ) f [nm] Error (r - x ) f [nm] Distane (x f x ) [mm] 1 5 PIDleadlag PIDSMC PIDleadlag PIDSMC PIDleadlag PIDSMC Time [s] Fig. 9. Slow positioning of DSA. The first graph shows a referene. The seond and third show a position error when a lead-lag ompensator and SMC are used for the oarse atuation, respetively. The last ompares the relative distane between the two atuators measured by the interferometers. with a maximum veloity of 37.5 mm/s, and the results are shown in Fig. 9. The DSA position error r x f is less than ±1 nm even during motion for the high disturbane rejetion of the PID, and it does not show major differene between the two DSA systems. However, the relative distane between the two atuators measured by the two interferometers (i.e. x f x ) is redued by almost 8% by the use of SMC. The graph also shows that the relative distane of the benhmark ontrollers (PIDleadlag) is lose to the stroke limit of the fine atuator (±1 mm). The seond experiment is fast positioning with a maximum veloity of.34 m/s, whih is 85% of the detetable veloity of the interferometers. Beause the fine atuator reahes its stroke limit with the benhmark ontrollers, this experiment is onduted only for the DSA with the PID and SMC. The results are shown in Fig. 1, where the relative distane stays well within the ±1 mm positioning range of the fine atuator. The settling time of the positioning over 1 mm to reah the ±1 nm error band is only.6 s. The position error is well suppressed within the range of ± 2.5 nm before and after the positioning without an external vibration isolation. 6554

6 Cape Town, South Afria. August 24-29, 214 Referene r [mm] Error (r - x ) f [nm] Error (r - x f ) [nm] Distane (x f -x ) [mm] Time [s] Fig. 1. Fast positioning of DSA using tamed PID and SMC. The first graph shows a referene. The seond and third show a position error with different sales. The last shows the relative distane between the two atuators measured by the interferometers 6. CONCLUSION In this paper a tamed PID and SMC have been individually designed and implemented for the fine and oarse atuators, respetively, to overome the range and speed limitation due to frition and external disturbanes in a DSA system. The experimental results show that the SMC of the oarse atuator redues the relative distane between the fine and oarse atuator by almost 8% during motion as ompared to the lead-lag ontrolled oarse atuator, whih prevents the fine atuator from reahing its range limits during repositioning. As a result, the DSA ontrolled by the tamed PID and SMC is able to ahieve a higher positioning speed while maintaining high preision without an additional vibration isolation system. ACKNOWLEDGEMENTS This work has been supported in part by the Austrian Researh Promotion Ageny (FFG) under projet number REFERENCES A. Al Mamun, I. Mareels, T. H. Lee, and A. Tay. Dual stage atuator ontrol in hard disk drive - a review. In The 29th Annual Conferene of the IEEE Industrial Eletronis Soiety, volume 3, pages , 23. B. Armstrong-Hélouvry, P. Dupont, and C. Canudas de Wit. A survey of models, analysis tools and ompensation methods for the ontrol of mahines with frition. Automatia, 3(7): , H. Butler. Position ontrol in lithographi equipment. IEEE Control Systems Magazine, 31(5):28 47, 211. A.H. Chaghajerdi. Sensing and ontrol in optial drives. IEEE Control Systems Magazine, 28(3):23 29, 28. L. Guo, D. Martin, and D. Brunnett. Dual-stage atuator servo ontrol for high density disk drives. In IEEE/ASME International Conferene on Advaned Intelligent Mehatronis, pages , G. Herrmann, C. Edwards, B. Hredzak, V. Venkataramanan, and S.K. Spurgeon. Appliation of a disrete sliding mode tehnique to a hdd dual-stage trak-seek and trak-following servo system. In IEEE 22nd International Symposium on Intelligent Control., pages , 27. X.P. Hu, S. Weerasooriya, T.S. Low, and B. M. Chen. Dual stage atuator modeling and ontrol in a d-rom drive. In Proeedings of the 24th Annual Conferene of the IEEE Industrial Eletronis Soiety, 1998., volume 3, pages vol.3, S. Ito, J. Steininger, P. I. Chang, and G. Shitter. Highpreision positioning system using a low-stiffness dual stage atuator. In 6th IFAC Symposium on Mehatroni Systems, 213. T. Ito and K. Nonami. Sliding mode ontrol with frequeny shaping to suppress spillover (in Japanese). Transations of Japan Soiety of Mehanial Engineers, 63(611):12 126, K. Kong, H.C. Kniep, and M. Tomizuka. Output saturation in eletri motor systems: identifiation and ontroller design. Journal of Dynami Systems, Measurement and Control, 132:512, 21. C.J. Lin, C.H. Wu, and C.C. Hwang. Traking ontrol of a motor-piezo XY gantry using a dual servo loop based on ILC and GA. In IEEE International Conferene on Control and Automation, pages , de. 29. R. Munnig Shmidt, G. Shitter, and J. van Eijk. The Design of High Performane Mehatronis. Delft University Press, 211. R. Nagamune, Xinghui Huang, and R. Horowitz. Robust H2 synthesis for dual-stage multi-sensing trak-following servo systems in HDDs. In Amerian Control Conferene, 26, pages , 26. T. Semba, T. Hirano, J. Hong, and L.-S. Fan. Dual-stage servo ontroller for HDD using MEMS miroatuator. IEEE Transations on Magnetis, 35(5): , Y. Shtessel, C. Edwards, L. Fridman, and A. Levant. Sliding Mode Control and Observation. Control Engineering. Birkhäuser, 213. Tsang-Li Tai and Jian-Shiang Chen. Disrete-time slidingmode ontroller for dual-stage systems - a hierarhial approah. Mehatronis, 15(8): , 25. T. Yamaguhi, M. Hirata, and C.K. Pang. High-Speed Preision Motion Control. Taylor & Franis,