A Review of Feedforward Control Approaches in Nanopositioning for High-Speed SPM

Transcription

1 Garrett M. Clayton Department of Mechanical Engineering, Villanova University, Villanova, PA Szuchi Tien Department of Mechanical Engineering, National Cheng Kung University, Tainan 71, Taiwan Kam K. Leang Department of Mechanical Engineering, University of Nevaa, Reno, NV Qingze Zou Department of Mechanical Engineering, Iowa State University, Ames, IA Santosh Devasia 1 Department of Mechanical Engineering, University of Washington, Seattle, WA evasia@u.washington.eu A Review of Feeforwar Control Approaches in Nanopositioning for High-Spee SPM Control can enable high-banwith nanopositioning neee to increase the operating spee of scanning probe microscopes (SPMs). High-spee SPMs can substantially impact the throughput of a wie range of emerging nanosciences an nanotechnologies. In particular, inversion-base control can fin the feeforwar input neee to account for the positioning ynamics an, thus, achieve the require precision an banwith. This article reviews inversion-base feeforwar approaches use for high-spee SPMs such as optimal inversion that accounts for moel uncertainty an inversion-base iterative control for repetitive applications. The article establishes connections to other existing methos such as zero-phase-error-tracking feeforwar an robust feeforwar. Aitionally, the article reviews the use of feeforwar in emerging applications such as SPM-base nanoscale combinatorial-science stuies, image-base control for subnanometer-scale stuies, an imaging of large soft biosamples with SPMs. DOI: / Introuction Control is critical for achieving high-banwith nanopositioning to increase the operating spee of scanning probe microscopes SPMs, such as scanning tunneling microscopes STMs 1 an atomic force microscopes AFMs. Increasing the operating spee of SPMs can significantly impact the throughput of a wie range of emerging nanosciences an nanotechnologies because SPMs are key enabling tools in the experimental investigation an manipulation of nanoscale biological, chemical, material, an physical processes; e.g., see Refs A critical problem in high-spee SPM operation is to position the SPM-probe precisely over the sample surface. Positioning errors between the SPMprobe an sample surface can lea to amage of the sample an/or the probe, as well as introuce unwante moification of the surface properties. To avoi the positioning problem at high operating spees inuce by mechanical vibration, commercial SPMs typically operate at low spees; for example, high-resolution SPMs scan the sample at aroun 1/1 to 1/1 of the lowest resonant-vibrational frequency. This positioning-relate limitation to SPM operating spee motivates the evelopment of control techniques that enable high-banwith nanopositioning in SPMs. This article reviews the role of feeforwar control in the evelopment of high-spee SPMs. While the SPM probe-to-sample istance coul be controlle using feeback methos, early SPMs i not have sensors to measure the scanning motion of the SPMprobe over the sample surface. Therefore, precision scanning of the probe over the sample surface relie on feeforwar control methos. For example, the inputs to the SPM-probe positioner 1 Corresponing author. Contribute by the Dynamic Systems, Measurement, an Control Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CON- TROL. Manuscript receive February 6, 8; final manuscript receive April 8, 9; publishe online October 8, 9. Assoc. Eitor: J. Karl Herick. usually, a piezoelectric actuator woul be cycle to reset hysteresis-memory effects an thereby achieve repeatable positioning. Such resetting of memory effects for SPM control is iscusse in Refs. 8,9. The avent of SPMs with sensors for measuring the probe s scanning motion 1 has substantially improve the SPM performance through the use of moern feeback methos 11. The aition of feeforwar control to these feeback schemes enables even further performance improvement in SPMs 1 by overcoming limitations impose by the nee to balance a high-banwith precision positioning with b robust close-loop stability in the presence of unmoele ynamics an uncertainties. This article reviews recent efforts in feeforwar control to increase the operating spee of SPMs. The article begins with a brief review of SPM operation an the nee for high-spee SPM to clarify the positioning issues in Sec.. Then, Sec. 3 iscusses the limits on SPM operating spee ue to vibration-cause positioning errors. Section 4 reviews feeforwar control methos for high-spee SPM along with current research an implementation issues. While this article oes not focus on SPM feeback control, it iscusses the integration of feeforwar with sensor-base feeback in Sec. 4, as well as an image-base approach to feeforwar control when sensors for feeback are not available in Sec. 5. Aitionally, Sec. 5 presents emerging applications where SPM feeforwar control plays an important role: a SPM-base nanoscale combinatorial-science stuies an b imaging of large soft samples with SPMs. A summary of the iscussions an concluing remarks is provie in Sec. 6. The Nee for High-Banwith Nanopositioning This section begins with a brief review of SPM operation to clarify the nee for high-banwith nanopositioning in SPM..1 SPM Operation: AFM Example. The ifference between various types of SPMs arises from the type of interaction, Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 / Copyright 9 by ASME

2 z,ref x ref y ref z z-axis Feeback Scan axes Controller x,y U z U x U y Piezoscanner z x y z Inuctive Sensors Laser Optical Sensor z Sample zs z zs Image x,y Fig. 1 AFM imaging in contact moe. The AFM probe is positione using a piezoscanner in the lateral scan x, y an vertical z axes. Other possible positioning schemes inclue the sample being move rather than the AFM probe an the use of separate scanners or multiple stages for ifferent axes. between the probe an the sample surface, that is measure or controlle. These inclue chemical, mechanical, electrical, an magnetic interactions. Moreover, SPM operation is ifferentiate by whether the probe-sample interaction is kept constant e.g., in contact moe or varie e.g., in tapping moe or uring nanofabrication 3. To illustrate typical SPM operation, consier contact-moe imaging using an AFM illustrate in Fig. 1, where the force between the AFM probe an the sample i.e., the probesample interaction is maintaine close to a esire value while the sample is scanne relative to the probe. Common to all SPMs is the nee to scan/position a probe over the sample surface. For example, in contact-moe AFM imaging, the AFM probe the tip of an AFM cantilever is scanne in a raster pattern across the sample surface by using a piezoscanner as illustrate in Fig. 1. While scanning the sample surface, the applie probe-sample force is estimate by measuring the eflection z of the AFM probe 13. It is note that the probe-sample force epens on the relative vertical position of the AFM probe with respect to the sample surface. Therefore, to maintain a constant probe-sample force, the measure AFM-probe eflection is fe back to generate an input U z that ajusts the vertical position z of the piezoscanner. This ajustment aims to maintain the AFMprobe eflection z at the esire value z,ref. Then, an AFM image of the sample topography is obtaine, in this case, by plotting the vertical position z s of the AFM-probe s tip over the sample against the lateral x-y position, as illustrate in Fig. 1.. The Nee for High-Spee SPM. SPMs can be use to a image an b manipulate ynamic surface phenomena at the nanoscale. For imaging an manipulating rapi ynamic processes, high-spee SPMs are neee, as escribe below...1 High-Spee Imaging. Distortion in the SPM-image occurs if the surface property being investigate changes rapily in time in comparison to the SPM s operating spee. The image istortion arises because measurements at the initial pixel an at the final pixel of an image are acquire at significantly ifferent time instants when the SPM-probe scans over the sample. Therefore, a high-spee SPM is neee to minimize the istortion for stuying, manipulating, an controlling processes with fast ynamics. For example, increases in the SPM s operating spee will avance the iscovery an unerstaning of ynamic phenomena by enabling a the stuy of rapi melting an crystallization of polymers ; b the investigation of fast phase transitions in ferroelectric materials 7 that influences omain formation, which in turn affects physical properties e.g., piezoelectricity, electrooptical properties, an hysteresis ; an c single-molecule vibrational an force spectroscopies to eluciate structural an electronic information 17,18... High-Spee Nanofabrication. The main avantage of SPM-base nanofabrication is that it achieves high-resolution atomic-scale features 5. Unfortunately, SPM-base nanofabrication suffers from throughput limitations that are present in all serial techniques the SPM-probe must visit each point where operation is neee. Even with multiple probes 19,, such serial processes cannot compete with parallel techniques such as optical lithography, which can process an entire wafer more precisely, one ie in one step. A solution to the low-throughput problem is to integrate the slower, top-own, SPM nanofabrication with faster, bottom-up, nanofabrication methos. For instance, rather than aing all the require materials in a irect write approach, STM-base chemical vapor eposition CVD might be use only for seeing or prenucleating the esire pattern, whereas the rest of the material can then be grown by selective CVD 1. Similarly, patterne self-assemble monolayers can be fabricate with AFM-base ip-pen nanolithography, which can then be use for nucleation an growth of functional polymers,3. In this sense, the top-own SPM is only neee for generating the initial pattern, which then forms the basis for growing the nanostructure using highly parallel bottom-up techniques 4,5. The SPM-base fabrication of the initial see pattern is faster than fabrication of the entire features; however, it is still slower than optical lithography methos. Therefore, high-spee SPMs are require to increase the throughput of emerging SPMbase nanofabrication techniques..3 The Nee for Nanopositioning in SPMs. Nanopositioning is critical in SPM applications. Broaly, two types of positioning are neee: a lateral positioning in the scan x-y axes see Fig. 1 an b vertical positioning along the z axis..3.1 Importance of Precision Lateral (x an y) Positioning. Precision lateral positioning is important when manipulating/ moifying the surface at a specific location on the sample, e.g., uring nanofabrication. For example, the SPM-probe nees to move along specifie scan trajectories x ref an y ref see Fig. 1 where surface manipulation is require uring nanofabrication 5. Lateral positioning errors lea to istortion of the achieve nanoscale features; therefore, nanopositioning is neee uring nanofabrication. In contrast to nanofabrication applications, lateral position is not critical in imaging applications, but esirable. In particular, uring imaging applications, precision x an y positioning is esirable to: a b c ensure that the esire image area size is achieve attain a uniform scan spee over the sample by using a triangular time trajectory for scanning to control scanspee-epenent effects in the measurements 6 scanspee variations cannot be avoie completely because acceleration is prevalent in turnarouns of the trajectory enable uniform spatial resolution across the image when using uniform time-sampling of the ata along with a triangular scan trajectory / Vol. 131, NOVEMBER 9 Transactions of the ASME

3 Sample (a) Probe follows sample profile Piezo Probe path Piezo Probe Sample (b) Probe oes not follow sample profile Fig. Precision vertical positioning allows the AFM probe to follow the sample profile plot a along each scan line without excessive probe eflection an, therefore, without excessive probe-sample force. If the AFM probe oes not follow the sample profile plot b, then the probe can ig into the sample, causing excessive probe eflection, thereby resulting in excessive probe-sample forces. avoi exciting oscillations in the vertical z-axis position of the SPM-probe cause by unwante oscillations in lateral position ue to the coupling between the lateral an vertical ynamics 7. An avantage of imaging applications over nanofabrication applications is that positioning errors in the scan irection can be alleviate by measuring the achieve x an y positions an then using the measure values, rather than the reference trajectories x ref an y ref, to plot the images; the use of reference trajectories was common practice before sensors were available in SPMs for measuring the lateral x an y positions 1. Thus, positioning error e.g., x x ref can be correcte in imaging applications by using position measurements. However, position measurements cannot be use to correct istorte surface features, ue to positioning errors, in fabrication processes..3. Importance of Precision Vertical z Positioning. Vertical positioning is critical for both imaging an manipulation/ moification of samples. For example, uring contact-moe AFM imaging, the vertical z-axis position, relative to the sample surface, affects the probe-sample force. Ieally, if the AFM-probe s position z s precisely follows the sample s topography, i.e., the sample profile along each scan line as shown in Fig. a, then the AFM-probe eflection an the probe-sample force can be zero. However, it is ifficult if not impossible to maintain zero, probesample forces in practice. In particular, probe-sample forces ten to be nonzero uring the initial approach to the sample when the probe-sample force transitions from a repulsive to an attractive regime, an the probe snaps into the surface. Moreover, a nonzero, probe-sample force is neee uring scanning to maintain contact between the AFM probe an the sample in the presence of isturbances such as thermal noise. Nonetheless, it is important to track the sample profile with nano- or subnano- scale precision. If the SPM-probe s tracking of the sample profile contains large errors, as illustrate in Fig. b, then the resulting excessive probe-sample force can cause large sample eformation in soft samples the original sample topography profile is then substantially ifferent from the AFM-probe position z s, as well as sample moification, an, possibly, sample amage. Even when imaging relatively har samples where sample amage is not a significant concern vertical positioning to track the sample profile is still important because excessive probe-sample forces ue to large positioning errors coul amage the AFM probe. Vertical nanopositioning is also critical when moifying the surface. For example, nanoscale features can be fabricate by using the AFM probe as an electroe to inuce local-oxiation through an electrical fiel applie between the AFM probe an the surface 8. The vertical position of the AFM probe with respect to the sample has a ominant effect on the applie current an the formation of the current-inuce oxie, i.e., on the features size an shape of the nanofabricate parts 9. Therefore, precision vertical positioning is important to control the probe-sample interaction being use to create the nanofeatures an, thereby, to avoi feature efects. 3 Limits to SPM Operating Spee The scanning motion tens to introuce vibration-cause error in the SPM-probe positioning, which, in turn, limits the operating spee of SPMs. 3.1 Why Is Position Control in SPM Difficult? Precision SPM-probe positioning is challenging because piezoscanners not only share typical positioning issues with other piezoactuatorbase nanopositioners, but also have large moel uncertainties relate to variations in the operating conitions. Typical nanopositioning problems with piezoactuators. SPMprobe positioning with a piezoscanner nees to account for a creep, b vibration, an c hysteresis 3. For a recent review, on typical nanopositioning issues, see Ref. 11. The vibrational ynamics see Fig. 3 a from Ref. 31 ten to have a low gain margin because of the rapi phase-rop associate with the small structural amping i.e., sharp resonant peak that is compoune by the effects of higher-frequency ynamics. Moreover, nonlinearities such as hysteresis effects see Fig. 3 b a to the challenge of esigning nanopositioning controllers. Problems specific to SPM. SPM-probe positioning shoul account for substantial uncertainty in the positioning ynamics ue to varying contact conitions between the probe an the sample 3,33, as well as the ifficulty in moeling phenomena such as meniscus interactions an viscous amping when operating SPM probes in liqui environments 34. Moreover, SPM-probe positioning nees to account for significant coupling between positioning along ifferent axes. For example, lateral positioning can significantly affect the positioning 7,35, as well as sensing 36,37, in the vertical irection. This is because oscillations in lateral position appear as extraneous variations in the sample topography that nee to be tracke in the vertical irection; there- Magnitue (b) Phase (egree) Frequency (Hz) (a) Frequency Response lateral isplacement (µm) µm Amplifier Input (V) (b)hysteresis Nonlinearity 1 Fig. 3 Experimental a frequency response with sharp resonances an b hysteresis nonlinearity for a tube-type piezoscanner. The input is the applie voltage to a piezoscanner preamplifier an the output is the lateral isplacement. Aitional etails are provie in Ref. 31. Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

4 Scan Frequency (khz) Barrett & Quate 1991 [39].1 * Resonant Vibrational Frequency Schitter et. al. 7 [4] Kuipers et. al [4] Rost et. al. 5 [44] Nakakura et. al [41] Ano et. al. 1 [43] Li & Bechhoefer 7 [38] Lowest Resonant-Vibrational Frequency(kHz) Fig. 4 Scan frequency is aroun 1/1 to 1/1 of the lowest resonant-vibrational frequency for a variety of SPM positioning systems an ifferent positioning controllers fore, lateral an vertical positioning are couple even when separate positioners are use for the lateral an vertical positioning. Overall, low gain margins, nonlinearities, coupling effects, an moel uncertainties make precision positioning of the SPM-probe challenging uring high-spee SPM operation. Scan Frequency (Hz) Ano et. al., 1 [43] Uchihashi et. al., 6 [53] Picco et. al. 7 [5] Humphris et. al. 5 [51] Rost et. al. 5 [44] Schitter et. al. 7 [4] ω sf =11.9 S 1. Viani et. al., ω sf = [47] S 1.6 Cupere et. al., 3 [54] Jiao et. al., 4 [55] 1 1 Ushiki et. al., [57] Scan Size(µm) Dong et. al., 3 [56] Barrett & Quate 1991 [39] Croft et. al. 1 [3] Ushiki et. al., [57] Fig. 5 SPM scan frequency sf versus scan size S with a variety of positioning systems operating in air represente by soli triangles, an in liqui over soft samples represente by. The lines are linear least-square-error fits of the ata points, soli line for air imaging, an ashe line for liqui imaging. When not reporte, the scan frequency an scan size were estimate if possible from the images, frame rates, an number of pixels Resonant-Vibrational Frequency Limits Scan Frequency. Positioning error cause by motion-inuce vibration in the SPM-probe positioner e.g., AFM piezoscanner in Fig. is a significant limitation to high-spee operation of SPMs. As the scan frequency is increase, relative to the lowest resonantvibrational frequency of the SPM-probe positioner, the vibrational moes of the positioner are excite, resulting in a loss of positioning precision. The vibration-cause positioning error tens to increase with the scan frequency because the major frequency components in the scan trajectory increase towar the resonantvibrational frequencies of the positioning system. For example, a triangular scan trajectory tens to excite the positioner vibration at a lower scan frequency, when compare with a sinusoial scan trajectory, because the triangular scan trajectory contains highfrequency harmonics of the scan frequency, which are not present in a sinusoial scan trajectory. The increase in the vibrationcause positioning error with the scan frequency limits the maximum acceptable scan frequency an, thereby, limits the maximum SPM operating spee. Typically, epening on the resolution neee, the achieve scan frequency is 1/1 to 1/1 of the lowest resonant-vibrational frequency. For example, the plot of SPM scanning frequency versus first resonant-vibrational frequency is shown in Fig. 4 for a variety of SPM positioning systems an ifferent positioning controllers As seen in the figure, the lateral scanning frequency is approximately 1% of the lowest resonant-vibrational frequency. Other vibration effects. The positioning error cause by exciting the positioner vibration is ifferent from those cause by vibration transmitte to the SPM from external sources. External vibration problems can be aresse using vibration-isolation schemes 45. Another potential source of errors is vibration in the SPM-probe; e.g., when low stiffness soft cantilevers are use to image/manipulate soft samples with an AFM. However, recent probe esigns reuce mass an stiffness simultaneously 46 an, thereby, achieve high resonant-vibrational frequencies in soft SPM-probes. The resonant-vibrational frequencies of these new probes are greater than 1 khz 47, which ten to be substantially higher than the SPM-probe positioner s resonant-vibrational frequencies in typical SPMs. Therefore, other vibration effects such as external vibrations an probe softness are not the major limitations to increasing the SPM s operating spee at present. 3.3 Increasing the Resonant-Vibrational Frequency. Since the piezoscanner s lowest, resonant-vibrational frequency limits the achievable scan frequency, an approach to enable nanopositioning at high scan frequencies is to increase the resonantvibrational frequencies of the piezoscanner. Using stiffer piezoplates 48 or miniaturize SPMs can increase the scan frequency because the resonant-vibrational frequencies of smaller an therefore stiffer piezoscanners ten to be higher,4 49. For example, the first bening resonant-vibrational frequency 1, for an unloae tube-type piezoscanner, is given by 1 = EI L A = E D D h 4L 1 where is the mass ensity, A= D D h /4 is the crosssectional area, D is the outsie iameter of the tube, h is the tube thickness, I= D 4 D h 4 / 4 16 is the cross-sectional area moment of inertia, L is the tube length, an E is Young s moulus. Note that the resonant-vibrational frequency 1 is inversely proportional to the square of the piezoscanner length. Therefore, as the piezoscanner length reuces, the scan frequency an the SPM operating spee increase. Decrease in scan range with smaller piezo. A consequence of increasing the SPM spee scan frequency by using smaller piezoscanners is that the scan range R of the piezoscanner reuces with the piezoscanner length L. For example, the maximum lateral isplacement range R of a tube-type piezoscanner is given by 5 R = 31 L v max D h where 31 is the piezoelectric strain constant an v max /h is the maximum, applie electric fiel, which is limite by material properties. Thus, the scan range ecreases as the piezoscanner is shortene. Traeoff between scan frequency an scan range. Eliminating the length L from the above two equations yiels v max 1 1 =.8 31 h R E 1 1 h/d 1 3 R Therefore, the first resonant-vibrational frequency 1 is inversely proportional to the scan range R. Similar to this relationship between the first resonant-vibrational frequency 1 an the scan range R in Eq. 3, there is an inverse relationship between the achievable SPM scan frequency sf an scan size S in reporte SPM systems, as shown in Fig. 5. In general, the achieve scan / Vol. 131, NOVEMBER 9 Transactions of the ASME

5 V H-1 (a) Inverse Hysteresis Hysteresis Nonlinearity H Piezoscanner Linear Dynamics G Output P V Hysteresis Linear Nonlinearity Dynamics Output - High Gain Notch Filter H G P (b) High-gain Feeback Control Piezoscanner V (c) Charge Control Charge Amplifier Hysteresis Nonlinearity H Piezoscanner Linear Dynamics G Output P Fig. 6 Three ifferent approaches to linearize hysteresis nonlinearity in piezoscanners frequency sf tens to be lower than the first resonant-vibrational frequency 1, an the achieve scan size S tens to be smaller than the maximum scan range R. A linear least-square-error fit of the ata in Fig. 5, which represents a variety of SPM-probe positioning systems, yiels sf 1/S 1.6 for imaging in air an sf 1/S 1. for imaging in liqui 4 This inverse relationship ictates the traeoff between the achieve SPM scan size S an the associate maximum scan frequency sf. Optimizing the traeoff between scan frequency an scan size. Recent efforts on nanopositioning aim to optimize the traeoff between scan frequency an scan range. The approach is to augment the isplacement of piezoactuators by using flexural-stage esigns. Such esigns aim to amplify the positioning range while minimizing the reuction in the resonant-vibrational frequencies 4,58 6. The systems in Refs. 6 6 have not been applie to SPM operations yet, scan frequency of the SPM-images was not reporte in Ref. 58, an the system in Ref. 59 was relatively slow since it was not optimize for high-spee operation; therefore, they are not inclue in Figs. 4 an 5. It is expecte that these esign approaches will increase the SPM scan frequencies for a given scan size in the future. Nevertheless, vibrational resonance is still present in these nanopositioning systems; reucing the vibration-cause positioning error can increase the SPM operating spee further. 4 Use of Feeforwar Approach in High-Spee SPM The central iea is that feeforwar inputs to achieve nanopositioning in high-spee SPM can be foun by inverting the piezoscanner ynamics, as first emonstrate in Refs. 1,3. To illustrate the approach, let G be the map between the input voltage V an the output position P of the piezoscanner P = G V 5 Then, to track precisely a given time trajectory of the esire output position P, the feeforwar input V ff can be foun by inverting the input-output map G as V ff = G 1 P 6 With choice in position trajectory. When there is flexibility in the choice of position trajectory P, e.g., the lateral x an y trajectories use in SPM imaging, other feeforwar methos, besies inverse feeforwar, can be use. Such feeforwar methos inclue a the input-shaping approach to minimize excitation of piezoscanner vibrational moes 63, b the optimization of scan trajectories to achieve a constant spee scan in one irection an minimize the retrace time 64, an c the reuction of the energy of the input signal at high frequencies to account for actuator limitations 65. Without choice in position trajectory. When there is no flexibility in the choice of position trajectory P, e.g., the lateral x an y trajectories use to fabricate specifie features with SPM-base nanofabrication, then the inverse input in Eq. 6 becomes the ieal feeforwar input because it leas to perfect tracking in the absence of external perturbations an moeling errors. The challenges, then, are to aress errors in the moel use to fin the inverse feeforwar V ff, an to integrate the feeforwar input with feeback to hanle external perturbations. These issues are iscusse in this section. 4.1 Linearizing the Hysteresis Nonlinearity Before Inversion. The hysteresis nonlinearity is typically linearize see ifferent schemes in Fig. 6 before using inversion as in Eq. 6 to obtain the feeforwar input 3. For example, inversionbase hysteresis compensation, see schematic in Fig. 6 a, can linearize the piezoscanner ynamics for SPM control as emonstrate in Refs. 3,66. It is note that inversion of the hysteresis nonlinearity, for general nanopositioners, has been stuie extensively in literature; e.g., see recent reviews in Refs. 11,67. Such inversion-base feeforwar can correct for hysteresis effects with high precision; the rawback is the substantial moeling complexity. An alternative approach without the moeling complexity is to reuce the hysteresis effects using high gain feeback. In this approach shown in Fig. 6 b, notch filters are use to increase the gain margin of the system as stuie in Ref. 68. Higher gain margin allows the use of higher-gain feeback, which substantially reuces the hysteresis nonlinearity 69. A thir approach is to use charge amplifiers rather than voltage amplifiers to rive the piezoscanner, which reuce the hysteresis nonlinearity, as shown in Fig. 6 c 7 7. The avantages of the charge control approach, which has been successfully applie to SPM in Ref. 73, are that it eases implementation, oes not require position sensing an feeback, an obviates the nee for etaile hysteresis moeling an inversion. An example of a linearize piezoscanner using a charge amplifier is shown in Fig. 7. The hysteresis nonlinearity was reuce from.35 m to.6 m; compare Figs. 3 b an 7 b 31. The Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

6 Magnitue (b) Phase (egree) Frequency (Hz) 1 3 (a) Frequency Response of G lateral isplacement (µm) µm Amplifier Input (V) (b) ResiualHysteresis.5 Fig. 7 Experimental a frequency response G an b resiual hysteresis nonlinearity for the tube-type piezoscanner an output range of 4 m asin Fig. 3. The input is the voltage applie to the charge amplifier as in Fig. 6 c an the output is the lateral isplacement. Aitional etails are in Ref. 31. resulting linearize piezoscanner, with ynamics G from input V to output P with any of the hysteresis linearization approaches in Fig. 6, can be use with the integrate feeback/feeforwar schemes iscusse below. 4. Integration of Feeforwar an Feeback Controllers. Although this article oes not focus on feeback controller esign, it is note that feeback control has been an essential part of SPM evelopment. For example, integral controllers are very effective in maintaining the esire level of probe-sample interaction, particularly uring low-spee operation, as they can overcome both creep an hysteresis effects in the piezoscanners an the vibrational ynamics are not ominant at low frequencies. In this sense, traitional proportional-integral-erivative PID feeback controllers or a ouble integral for tracking a ramp are well suite for nanopositioning an, therefore, have become popular in lowspee SPM applications 1. Recent works have aime to robustify such traitional integral controllers in SPMs 74. Starting with the early work in Ref. 75, moern feeback control techniques 58,76 79 have been successfully applie to SPM. See Ref. 8 for a comparative review of ifferent feeback methos in SPM. Two approaches use to integrate feeforwar an feeback in SPM are illustrate in Fig. 8. These approaches, compare in Ref. 81, are briefly iscusse below. Close-loop inversion. In general, the moel-inversion-base feeforwar approach cannot correct for positioning errors ue to moel uncertainties 85. Nevertheless, feeback can be use to reuce the effects of piezoscanner uncertainty in the close-loop system G cl. Then, as shown in Fig. 8 a, the close-loop system G cl is inverte to obtain the feeforwar input V ff = G 1 cl P 7 This close-loop inversion approach that reuces the moel uncertainty before inversion reuces the computational error in the feeforwar input as emonstrate in Refs. 38,8,83. Moreover, both the feeback an the inverse feeforwar can be simultaneously optimize 81 to improve the positioning performance as emonstrate for SPM control in Refs. 86,87. Plant inversion. In the plant-inversion approach, shown in Fig. 8 b, the exact positioning feeforwar input V ff is the inverse of the piezoscanner ynamics G, as shown in Eq. 6. With this inverse input, V ff =G 1 P, the tracking error is zero in the absence of moeling errors an external perturbations such errors are correcte using feeback. An avantage of the plantinversion approach is that it tens to have better positioning performance than the close-loop-inversion approach. This is because the inverse feeforwar, with the plant-inversion approach, oes not share the performance limitations of the closeloop system, which arise because feeback controllers trae off positioning performance such as banwith to ensure stability uner plant uncertainties. Aitionally, the nonminimum-phase nature of typical piezoscanner ynamics, as well as input saturation bouns, tens to limit the performance of the close-loop system 58. The isavantage of the plant-inversion approach is that it cannot be use in the presence of large moel uncertainty, especially, when iterative proceures 7 are not applicable to reuce the uncertainty-cause error, e.g., lateral positioning in nonrepetitive nanofabrication. In such applications, the moel uncertainty nees to be reuce by ientifying the moel uner the specific operating conitions. 4.3 Inversion Approaches. This subsection reviews the use of the inversion-base approach to fin feeforwar inputs that correct for the piezoscanner ynamics an, thereby, enable nanopositioning of the SPM-probe. In the following, the plantinversion approach Eq. 6 is escribe. The proceure is similar for the close-loop inversion approach Eq. 7 in which the close-loop transfer function G cl is use instea of the plant transfer function G Piezoscanner Moel. To illustrate ifferent issues in inversion, consier the following moel G s of the linearize piezoscanner for positioning along a single axis lateral or vertical Desire Position P Feeforwar Controller V ff Close Loop System Gcl V Feeback - Controller G Output P Desire Position P - Feeforwar Controller Feeback Controller V ff V fb V G Output P (a) Close-loop Inversion, V -1 ff = G cl [ P ] (b) Plant Inversion, V -1 ff = G [ P ] Fig. 8 Integrating feeforwar with feeback in SPM: a Close-loop-inversion approach 38,8,83 an b plant-inversion approach 7, / Vol. 131, NOVEMBER 9 Transactions of the ASME

7 Table 1 Natural frequencies an amping ratios of linearize moel G in Eq. 8 G s = P s V s = K s z z s z s p1 p1 s p1 s p p s p 8 where the output P is in nanometers an the input V is in volts. In the following, K is chosen to be 1 1 an the other parameters are given in Table 1. With this choice of parameters, the c-gain G = ra/s Zeros z = 1 z =.1 First resonance poles p1 = 5 p1 =.5 Secon resonance poles p = 3 p =.5 K z p1 p becomes G =11.6 nm/v, which is a typical isplacementper-volt range for tube-type piezoscanners. For example, with a tube-type piezoscanner with L=1 cm, h=1 mm, D=1 cm, an 31 =.17 nm/v for PZT-5A material 88, Eq. yiels a lateral isplacement-per-volt of 153 nm/v. The secon vibrational frequency p is chosen to be approximately six times the first vibrational frequency p1, an the zero z is interlace between the poles to represent bening vibrations in beams. The frequency response of the example system with the above parameter values is shown in Fig. 9 a. The piezoscanner ynamics a is stable; b has zeros on the right half of the complex plane, i.e., is nonminimum phase, as in typical SPM positioning systems, e.g., Refs. 1,58 ; an c has relative egree of. The relative egree, the ifference between the orer of the transfer function s enominator an numerator, being implies that the input voltage V can irectly change the secon erivative of the output 89. Thus, in this case, the input irectly changes the acceleration of the output position P as in typical mechanical systems. Note that the use of filters to smooth the input or the output tens to increase the relative egree of the system c-gain Inverse. The simplest feeforwar metho is c-gain inversion, where the feeforwar input V ff t is foun as V ff t = G 1 P t 1 where P is the esire output position trajectory. This metho works aequately for slow esire trajectories, but results in significant vibration an positioning error as the operating frequency is increase because the ynamics are not accounte for. 9 Positioning error analysis. The positioning error e= P P with the c-gain approach increases with the frequency content, as well as the amplitue A p of the esire trajectory. The error ynamics can be obtaine from Eqs. 8 an 1 as e s = P s G s V ff s = 1 G P G s s = G e s P s 11 The steay-state error when the esire position is a sinusoial of frequency, say, P t = A p sin t 1 is given by e t = A p G e j sin t G e j 13 Then, the maximum positioning error e max an the percent maximum positioning error %e max, at the steay state, are given by e max = A p G e j %e max = 1e max /A p = 1 G e j 14 where G j an %e max are shown for ifferent frequencies in Fig. 9. Note that the maximum positioning error e max increases with both a the amplitue of the position trajectory A p, as well as b the frequency that nees to be tracke. Therefore, for the same amount of acceptable positioning error e max, the maximum positioning frequency max reuces as the amplitue of the position trajectory A p increases. For example, with an acceptable maximum error e max = nm an amplitue A p =1 m, the maximum percentage error nees to be less than %, which occurs at 43.3 Hz in Fig. 9 b this is slightly less than one-tenth of the first resonant-vibrational frequency at 5 Hz. If the positioning amplitue is increase ten times to A p =1 m, then the maximum percentage error nees to be less than.% an the maximum scan frequency is reuce to 4.54 Hz. Nevertheless, the c-gain approach can be use when the position trajectories frequency components are sufficiently low in comparison to the resonantvibrational frequencies of the piezoscanner Inverting Perioic Trajectories. When the position trajectory is perioic, e.g., for lateral scanning, the computation of the inverse in Eq. 18 can be simplifie using steay-state solutions as in Refs. 83,9,91. In particular, let the esire position P be perioic; i.e., P t = A k sin k t k 15 k Then, the inverse feeforwar in Eq. 6 can be compute as 83 Magnitue of G (b) (a) Frequency (Hz) Percent error % e max Frequency 1 (Hz) (b) Fig. 9 a Magnitue response of example system G in Eq. 8 an b percentage error %e max in Eq. 14 for a sinusoial position trajectory. The percentage error reaches % at 43.3 Hz, which is slightly less than one-tenth of the first resonant-vibrational frequency at 5 Hz. Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

8 1 6 6 (a) V ff *1 (V) P (nm) time (s) output, P (nm) error, P P (nm) (b) time (s) Fig. 1 Inversion for perioic position trajectory. a Feeforwar input V ff an esire position P for system G in Eq. 8. b Simulate transient error P P in position P when the perioic inverse input V ff is applie to G. A V ff t = k k G j k sin kt k G x j k 16 For example, the feeforwar input V ff when the esire position P is the first five o harmonics of a 5 Hz triangular trajectory k = k 5, A k = k 1 / k, k = for k = 1,3,5,7,9 17 is shown in Fig. 1 a for the system G in Eq. 8. While the approach oes not aress transient vibration, the error in the resulting position ue to initial conition mismatch ecreases exponentially as shown in Fig. 1 b, because the piezoscanner ynamics G is stable. For this simulation, the positioning error reuces to less than nm after 3 cycles. In general, the time neee for the transient error to become sufficiently small epens on the poles of the system G s. Application of perioic-trajectory tracking in SPM. An avantage of this perioic-trajectory inversion is that for tracking a purely sinusoial trajectory of frequency, it requires the estimation of two parameters, G j an G j, which coul be ientifie online. This approach is well suite to lateral control in SPM for imaging applications, if images are acquire after the transient error becomes small as emonstrate in Ref. 9. The approach can also be aapte to ientify the SPM ynamics as shown in Ref. 91. Extension to repetitive control (RC). The formulation as the tracking of a perioic trajectory lens itself to techniques from repetitive control 9,93 that can be applie to nonminimumphase systems 94. In such applications, the perioic input in Eq. 16 can be foun by exploiting the internal moel principle 95, where asymptotic tracking of the esire output is achieve provie the generator for the esire output is inclue in the stable, close-loop system. An avantage of the repetitive control approach is that it can be easily integrate into an existing feeback controller in SPMs to hanle tracking error associate with perioic motion an/or to reject perioic exogenous isturbances 96. Recently, the applicability of repetitive control methos to lateral positioning in SPM was emonstrate in Refs. 96, Exact Inverse as Feeforwar. If the linearize ynamics is minimum phase no zeros on the open right half of the complex plane, then the inverse feeforwar can be foun by inverting the system ynamics 98 V ff s = G 1 s P s 18 Inverse is not proper. An issue with the exact inverse in Eq. 18 is that the inverse G 1 s is not proper. The input voltage V irectly changes the rth time erivative of position P, where r is the relative egree the ifference between the orers of the enominator an numerator polynomials of the transfer function G 89. Therefore, the esire position P has to be sufficiently smooth ifferentiable r times with respect to time for exact tracking. For the example system Eq. 8, with relative egree r=, the esire position P must be twice ifferentiable in time to be tracke exactly. Let P t = /t P t ; then, the inverse in Eq. 18 can be rewritten as V ff s = G 1 s s s P s = s G s 1 P s = G # s 1 P s 19 where the inverse G # s 1 is proper since G # s =s G s is proper with relative egree zero. It is note that being twice ifferentiable is a necessary conition for exact tracking an a trajectory P t that is not twice ifferentiable in time cannot be tracke with finite inputs 89. Therefore, if the trajectory P can be tracke, then the nee to specify the secon time erivative P of the position P oes not impose an aitional requirement. Inverse for nonminimum-phase systems. The exact inverse G 1 s woul be unstable if the system G s is nonminimum phase since the right-half-plane zeros of G s become the unstable poles of G 1 s. Therefore, a stanar inverse Eq. 18 or Eq. 19 woul result in an unboune feeforwar input V ff over time for a general position trajectory P. In contrast, a boune although noncausal feeforwar input can be foun, for the nonminimum-phase case, using the Fourier-transform approach by Bayo in Ref. 99. For the example system Eq. 8, the exact inverse can be foun as V ff j = G # j 1 P j Then, the time-omain inverse input V inv is obtaine using the inverse Fourier transform. A time-omain interpretation was evelope in Ref. 1 an inverse for nonlinear nonminimumphase systems was evelope in Ref. 11. For the example system Eq. 8, the esire acceleration P, the esire position P, an the exact-inverse feeforwar V ff are shown in Fig. 11. The triangular sections of the esire acceleration P in Fig. 11 can be escribe using Eq. 17 with k =k 5 an scale to generate an output P with a maximum isplacement of 1 nm. Note that the inverse input V ff foun in the Fourier omain remains boune even though the inverse system G 1 s has unstable poles. Also, note that the inverse V ff is noncausal it is nonzero before the output acceleration P starts to change as seen in Fig. 11. Therefore, the feeforwar input V ff has to be applie a priori, before the output starts to change / Vol. 131, NOVEMBER 9 Transactions of the ASME

9 Input, esire output & acceleration V ff *1 (V) 8 P (nm) 1 P *1-6 (nm/s ) Time (s) Fig. 11 Noncausal, exact-inverse feeforwar V ff for a esire output acceleration P an the associate esire output trajectory P. Note that the inverse feeforwar V ff is nonzero before the output acceleration P starts to change. Exact inversion in the time omain. The time-omain approach to exact inversion is presente below. Let the system G be given in state space form as ẋ t = Ax t BV t P t = Cx t 1 In the following, expressions are provie for the general case of relative egree r; for the example, expressions can be obtaine by setting the relative egree as r=. The relative egree r implies that the input V appears in the expression for the rth time erivative of the output position P as 1 r t r P t = CAr x t CA r 1 BV ff t = A y x t B y V ff t where B y is guarantee to be invertible. The inverse system can be written as 98,1 = P t P = 1 C t t CA ] ] r 1 t t r 1 P t CA = T r 1 x t T x t T t t = Tx t 1 5 x t = T t = T t 1 l T 1 r t 6 t an the bottom portion T of the coorinate transformation matrix T in Eq. 5 is chosen such that T is invertible. For the example system G in Eq. 8, = 1 p1 p1 p1 A 1 p p p, = B 4, C = an with T in Eq. 5 chosen as T = the matrices associate with the inverse system in Eq. 3 are given by A inv = , B inv = , where t = A inv t B inv Y t V ff t = C inv t D inv Y t A inv ª T A BB y 1 A y T r 1 B inv ª T A BB y 1 A y T l 1 ] T BB y 1 C inv ª B y 1 A y T r 1 D inv ª B y 1 A y T l 1 ] B y 1 Y t ª t t r t r P an T represents a state transformation 3 4 C inv = , D inv = Remark 1. Unstable internal ynamics. The eigenvalues of A inv Eq. 3, i.e., the poles of the inverse system, correspon to the zeros of the original system G s Eq. 8. Therefore, the inverse system is unstable for nonminimum-phase systems. For general hyperbolic systems with no zeros on the imaginary axis, to fin a boune solution to the unstable inverse system Eq. 3, the internal state is ecouple into stable an unstable states using a coorinate transformation T split T split s t = t u t The new state equation is given by t s t = T u t 1 split A inv T split s t T u t 1 split B inv Y t = A s A u s t u t B s B u Y t 3 31 Remark. The transformation matrix T split is chosen to ecouple the internal state into stable an unstable subynamics. Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

10 The transformation matrix can be foun using the eigenvectors of the matrix A inv. Remark 3. For the example system Eq. 8, both the poles of A inv are unstable; therefore, the stable portion s of the internal ynamics is null. In general a boune solution to the ecouple internal state equations can be foun as t s t = A s s t B s Y t u t = A u u t B u Y t s,ref t e = A s t B s Y solve forwar in time u,ref t e = t A u t B u Y solve backwar in time 3 The reference internal state trajectory is then foun from Eqs. 3 an 3 as ref t = T split s,ref t 33 u,ref t an the inverse input can be foun from Eq. 3 as V ff t = C inv ref t D inv Y t 34 Remark 4. The nee for preview. Computing the boune solution to the unstable internal state u,ref at time t require information about future values of the esire output, e.g., Y with t in Eq. 3. The noncausality of the inverse input V ff implies that the future, esire output trajectory nees to be known ahea of time for computing the inverse, e.g., in the Fourier omain Eq. or in the time omain see Remark 4. This is acceptable for lateral positioning in SPM when the esire x an y trajectories are known ahea. Preview-base implementation of noncausal inverse. In some applications the entire esire position trajectory might not be available, but at any time instant t, future position information might be available for a finite preview-time interval t,tt p. For instance, the lateral position trajectory nees to be change in nonraster scanning use to image only parts of the scan area that is of interest 13. In such cases, the inverse feeforwar can be compute using the finite preview information of the esire position trajectory. In particular, the boune solution to the unstable internal state u,ref at time t only uses finite preview information if its boune solution in Eq. 3 is approximate as 1 u,ref t = ttt p e A u t B u Y 35 The amount of preview time T p neee can be quantifie in terms of the require tracking accuracy an the system ynamics 1. Such a preview-base approach was emonstrate for SPM control in Ref. 14. If sufficient preview time is not available, then, the feeforwar can be optimize for minimizing the tracking error for the available amount of preview 15,16. Causal approximate inverse for nonminimum-phase systems. In some applications, preview information of the output position may not be available, e.g., when tracking the vertical probesample istance to correct for surface variations uring SPM nanofabrication. For nonminimum-phase systems, the lack of preview in these applications is a major limitation to achieving precision positioning because the noncausal exact-inverse cannot be use. Causal controllers cannot perfectly track general trajectories when the system is nonminimum phase 17. The only recourse then is to use causal feeforwar approaches that can achieve asymptotic tracking for certain classes of trajectories 89,18, an approximate tracking for more general trajectories, e.g., Refs. 19,11. A review of approximate inversion is presente in Ref. 81. If the positioning error with the causal feeforwar is too large, then the SPM operating spee nees to be reuce so that the position error can be lowere to an acceptable level thus, limiting the operating spee. 4.4 Inverse Feeforwar Uner Moel Uncertainty. It can be shown that the aition of inverse feeforwar can improve the positioning performance when compare with the use of feeback alone, even in the presence of plant uncertainties the size of acceptable uncertainty is quantifie for general linear timeinvariant systems in Refs. 87,111. In particular, for single-input single-output systems, such as SPM-probe positioning along a single axis, performance improvement with the aition of feeforwar can be guarantee at frequencies where the uncertainty j in the nominal plant is smaller than the size of the nominal plant G j 111 as follows: j G j 36 Remark 5. Noncausality of inverse uner uncertainty. Typical systems ten to have large moel uncertainties at higher frequencies; the resulting inverse in frequency regions satisfying Eq. 36 is noncausal even if the system G is minimum phase 111. In particular, if the inverse is set to zero in any frequency interval of nonzero length, then the resulting time-omain input V is noncausal by the Paley Wiener conition Optimal Inverse. The acceptable uncertainty bouns, for guarantee performance improvement with the use of the inverse feeforwar in Eq. 36, are often violate in typical systems. Most piezoscanners ten to have some frequency regions where plant uncertainty is unacceptably large usually at high frequencies an near system zeros. Therefore, the inverse shoul be compute only in frequency regions where the plant uncertainty is sufficiently small. The optimal inverse evelope in Ref. 11 allows the inverse to be compute at specifie frequency ranges. The optimal inverse input is foun by minimizing the following cost function that is similar to the cost function use in stanar linear quaratic regulators with frequency-epenent weights as in Ref. 113 : J V = V j R j V j E P j Q j E P j 37 where enotes the complex conjugate transpose an E P = P P is the positioning error. The terms R j an Q j are realvalue, frequency-epenent weightings that penalize the size of the input V an the positioning error E P 113. Remark 6. Weights for large uncertainty. If the moel uncertainty is large at some frequency, then the choice of weights R j an Q j = results in a minimum cost with zero input V j =, i.e., no tracking at that frequency. Remark 7. Actuator reunancy an eficiency. The above cost function can be use to optimize the input with reunant actuators in multistage positioning systems, an for optimal tracking in actuator eficient systems 114. The optimal inverse input V opt that minimizes the cost function in Eq. 37 can be foun, for the single-input-single-output SISO case, as / Vol. 131, NOVEMBER 9 Transactions of the ASME

11 Magnitue (b) (a) system G*1 4 exact inverse G 1 1 optimal inverse G opt 1 ZPET G ZPET Frequency (Hz) Phase (egree) (b) system G*1 4 exact inverse G 1 optimal inverse ZPET 1 G ZPET o 1 G 1 o opt Frequency (Hz) Fig. 1 Comparison of exact inverse G 1, optimal inverse G 1 1 opt, an ZPET inverse G ZPET. a Magnitue an b phase. Note that the phase plots for the optimal inverse an ZPET inverse are shifte to istinguish them from the phase plot of the exact inverse. G V opt j = j Q j R j G j j Q j G j P = G 1 opt j P j 38 an the time-omain signal for the feeforwar input V ff t =V opt t is then obtaine through an inverse Fourier transform of V opt j. Time-omain computations using impulse response of G 1 opt, as well as a preview-base implementation, are iscusse in Ref. 14. Remark 8. Frequency-weighte inverse. The optimal inverse in Eq. 38 can be expresse as G 1 G opt j = R j G j Q j G j G 1 j = opt j G 1 j 39 Therefore, the optimal inverse can be consiere as a frequency weighte inverse Comparison of Optimal an Exact Inverses. The exact inverse tens to increase at high frequencies where the system G tens to have low gains, especially when the frequency is high in comparison to the first, resonant-vibrational frequency of the positioning system as shown in Fig. 1. A problem with such an inverse is that moeling uncertainties also ten to be large at higher frequencies. One approach is to suppress exciting the higher-frequency uncertain regions by using a relatively large weight on the input R at such frequencies in the cost function Eq. 37. The weights in the cost functions Q j an R j are real value when they have the form R j = r j r j, Q j = q j q j 4 For this example, r an q are chosen to suppress exciting frequencies beyon the main resonant-vibrational frequency at 5 Hz as w s = 5, q s = w s, r s =1/w s 41 s 5 The resulting optimal inverse G 1 opt in Eq. 38 an the exact inverse G 1 in Eq. are compare in Fig. 1, for the example system G Eq. 8, whose response is also shown in the same plot. The magnitue of the optimal inverse tens to reuce at higher frequencies by esign, to suppress exciting these regions as shown in Fig. 1. Thus, the optimal inverse enables traeoffs between the esire to track an output precisely an the nee to account for the moel uncertainty as well as actuator banwith limits Comparison of Optimal Inverse an ZPET Feeforwar. The optimal inverse achieves the appropriate phase correction at all frequencies as in zero-phase-error-tracking ZPET feeforwar 19. In particular, if the weights Q j an R j are chosen to be zero phase e.g., as in Eq. 4, then the optimal inverse in Eq. 38 has the same phase as the exact inverse since G = G 1, as shown in Fig. 1 b. Therefore, the optimal inverse has the appropriate phase correction, base on the nominal moel, at all frequencies. This is similar to the ZPET feeforwar in Ref. 19, where a minimum-phase system G ZPET is use to approximate the nonminimum-phase ynamics by replacing the nonminimum-phase zeros with stable poles before inverting to fin the feeforwar input. For the example system Eq. 8, the ZPET approximation results in G ZPET s = K z 4 s z z s z s p1 p1 s p1 s p p s p 4 where the aitional pair of poles has the same imaginary parts as the original zeros, but with a change in the sign in the real part. If at some frequency, the moel uncertainty is not too large for a nonminimum-phase system, then the optimal inverse provies the appropriate magnitue gain correction for the nominal plant, even at high frequencies. In contrast, the ZPET feeforwar aims to achieve goo magnitue gain correction in the low-frequency region an not at high frequencies as shown in Fig. 1 even if the moel uncertainty is zero ue to the replacement of the nonminimum-phase moel by a minimum-phase one 19. Thus, Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

12 both the optimal inverse an the ZPET feeforwar achieve zerophase-error tracking; however, the optimal inverse tens to achieve better performance than the ZPET feeforwar in highbanwith tracking Comparison of Optimal Inverse an Robust Feeforwar. The optimal inverse G 1 opt can be consiere as the noncausal generalization of H -robust feeforwar 115, where the feeforwar controller G ff is foun by minimizing the worstcase system energy gain 116 r G ff 43 q 1 G ff G over all causal, stable controllers G ff to obtain the feeforwar input as V ff s = G ff s P s 44 where r s an q s are frequency-epenent weights as in Eq. 4. It is note that for any feeforwar controller G ff, an any given esire output P, the integran I J in the cost function J V for the optimal inverse in Eq. 37 can be rewritten as using Eq. 4, an by substituting E p = P P I J V,P j = V j r j r j V j E P j q j q j E P j r j V j = 45 q j P j P j for all frequency, where a is the square of the stanar vector -norm for a C n. With the feeforwar controller G ff, substituting V=G ff P for the input an P = GV = GG ff P 46 for the output, the integran in Eq. 45 can be rewritten in terms of the feeforwar controller G ff as r j G I J G ff,p j = ff j P j 47 q j 1 G j G ff j P j As shown in Ref. 11, the optimal inverse V opt =G 1 opt P minimizes the cost function J V in Eq. 37 by minimizing the integran in Eq. 45 at each frequency for any esire output P. Therefore, I J G 1 opt,p j I J G ff,p j 48 at each frequency. This implies that the integral over the entire frequency omain shares the similar relationship as the integran is non-negative at each frequency J H G ff = I J G 1 opt,p j I J G ff,p j which, in turn, results in I J G 1 opt,p j I J G ff,p j sup sup P P P P 49 5 where P represents the square of the functional -norm. The above equation implies that the H -norms i.e., the inuce function norms are relate as r G 1 opt q 1 G G 1 opt r G ff q 1 G G ff G 51 Therefore, the optimal inverse minimizes the cost function J H G ff in the H -norm over all feeforwar controllers G ff causal or noncausal, i.e., y-scan irection (Angstroms) y-scan irection (Angstroms) x-scan irection (Angstroms) x-scan irection (Angstroms) (a) 445 Hz (Without Inverse) (b) 445 Hz (With Inverse) Fig. 13 Comparison of STM images of HOPG surface without plot a an with plot b the optimal inverse input 1. The istortion of the uniform lattice pattern of the sample in plot a is substantially reuce with the use of the optimal inverse in plot b. J H G 1 opt = inf J H G ff 5 G ff Thus, the H cost of the optimal inverse is less than or equal but not more than the H cost for the stanar, robust feeforwar controller obtaine by restricting the solution space to causal controllers Application of Optimal Inverse to SPM. The optimal inverse was applie to achieve high-spee imaging with a STM in Ref. 1. The use of the optimal inverse is comparatively evaluate without the use of the optimal inverse in Fig. 13, which shows STM images of a highly oriente pyrolytic graphite HOPG surface. Note that the uniform lattice pattern of the HOPG sample is istorte significantly ue to vibration-cause positioning errors in the scan trajectory Fig. 13 a. In contrast, the istortion can be reuce by using the optimal-inversion-base feeforwar input, an the image captures the expecte lattice pattern Fig. 13 b. Thus, positioning errors can be reuce with the optimal inverse to increase the SPM s operating spee IIC. When the positioning application is repetitive e.g., uring perioic scanning of the SPM-probe, iterative control methos can improve the positioning performance. Therefore, iterative 7,117 1 an aaptive control methos 13 are well suite for SPM applications 11. For example, uncertainty in the inversion process can be reuce using aaptive inversion of the system moel or by learning the correct inverse input that yiels perfect output tracking, i.e., iterative inversion of the system moel 7,14,1. The application of such iterative control to SPM was emonstrate in Refs. 7,61,91,15,16. Iterative approaches can also be use to reuce positioning errors ue to hysteresis as emonstrate in Refs. 8, IIC Proceure. The inverse G 1 was use in the iteration law as early as Ref. 18. The novelty in recent IIC is the use of the noncausal inverse G 1 in the iteration law 7,14. In particular, at the kth iteration step, the input is V ff,k j = V ff,k 1 j j G 1 j P j P k 1 j 53 where k 1, V ff, j is the initial input, j R is the frequency-epenent iteration-gain, an P j P k 1 j represents the positioning error at the previous k 1 th iteration step. Convergence of such proceures was stuie in Refs. 7,14,19. In particular, a frequency omain criterion was stuie in Ref. 7, which showe that the IIC proceure converges at a frequency provie the phase error in the moel is less than /, an the upate gain j is sufficiently small, i.e., / Vol. 131, NOVEMBER 9 Transactions of the ASME

13 j cos j an / 54 M j where M is the magnitue uncertainty an is the phase uncertainty. Similar requirements on the phase being less than / egree arise in time-omain approaches to prove convergence of IIC 19, as well as in repetitive control methos for perioic trajectories 94,96. Remark 9. Choosing the iteration-gain. The optimal inverse can be use to fin the frequency-epenent iteration-gain j in Eq. 53, which accommoates the moel-uncertainty variation with frequency, by choosing j = opt j as in Remark 8, Eq. 39. Remark 1. Noncausality of the IIC input. If there is substantial uncertainty at higher frequencies, then the iteration-gain j in Eq. 53 nees to be zero at those high frequencies to guarantee convergence, an, therefore, the IIC input in Eq. 53 is necessarily noncausal even for a minimum-phase system G as in Remark 5. Moel-less IIC. The convergence of the inversion-base iterative control IIC Eq. 53 is limite by the moeling error of the linear ynamics map G, particularly the phase. Such a moelingrelate constraint can be alleviate by aaptively upating the ynamics moel G along with the iteration. Towar this, the inverse G 1 of the ynamics use in the IIC algorithm Eq. 53 is obtaine from the measure input-output ata from the previous iteration step 61,9,16, i.e., G 1 k j V ff,k 1 j 55 P k 1 j provie P k 1 j. Moreover, since the confience level in the above online ientification process is high, the upate gain is chosen as 1. Then, the IIC algorithm Eq. 53 becomes 61,16 V ff,k j = V ff,k 1 j V ff,k 1 j P k 1 j P j P k 1 j = V ff,k 1 j P k 1 j P j 56 provie P k j an P j at frequency ; the input V ff,k j = otherwise. The approach has been shown to be effective even with an initial moel G=1. When sensors are not available to measure the actual output P, the input-output ata neee to fin the moel in Eq. 55 can be obtaine uring the iteration process by using an image-base approach 9,91. The challenge is to prove convergence in the presence of noise an isturbances that affect the estimate moels. Convergence issues are stuie in Ref. 16. When compare with IIC with a fixe nominal moel, the ientification of the moel uring iterations has emonstrate substantial improvement in the convergence rate for nanopositioning applications in Refs. 61, Iterative Control to Correct for Nonlinearities. Iterative approaches can be use to reuce positioning errors ue to hysteresis, as emonstrate for SPM in Refs. 8,17. While linearization methos e.g., shown in Fig. 6 o reuce the hysteresis effect, iterations can reuce the resiual hysteresis further, especially, uring large-range positioning. For example, the resiual hysteresis can still be as large as 6 nm when the position changes by 4 m as shown in Fig. 7. Convergence in the presence of hysteresis. The major challenge is to emonstrate convergence of iterative algorithms in the presence of hysteresis. The ifficulty in proving convergence arises ue to branching effects, which prevent the iterative algorithm from preicting the irection in which the input nees to be change base on a measure output error. This is similar to the phase uncertainty being larger than / in traitional iterative control methos. To illustrate this loss in irection ue to branching effects, consier a typical hysteresis output P versus input Fig. 14 Hysteresis curve with two branches: ascening B a an escening B V response shown in Fig. 14. For a esire output P, there are two possible input values, V H on the ascening branch B a an V L on the escening branch B. Then given an input-output pair V, P an, therefore, the output error P P, it is unclear if the input shoul be increase to V H or ecrease to V L unless it is known that the system is on the ascening branch. Therefore, convergence is establishe branch by branch as shown in Refs. 8,67,13 for iterative proceures of the form V ff,k t = V ff,k 1 t H1 P t P t 57 Inverse-hysteresis iterative control. In aition to branching effects, the convergence of iterative approaches is also affecte by the nonlinearity of the hysteresis. If the nonlinearity is not accounte for as in Refs. 8,13, then the iteration-gains nee to be substantially small to enable convergence. Recent work 131 inclues an inverse hysteresis operator H 1 in the iteration proceure, as V ff,k t = V ff,k 1 t H H 1 P t H 1 P t 58 Convergence issues are stuie in Ref. 131 to show that inverting the hysteresis nonlinearity can substantially improve the convergence rate of iterative algorithms. Moreover, this approach was use to improve lateral positioning precision an, thereby, feature accuracy in AFM-base nanofabrication in Ref Application of IIC to SPM. IIC yiels the highest precision in SPM positioning because of its ability to correct for moeling errors. For example, reuction of positioning error close to the sensor noise level was emonstrate in Refs. 7,61,16. This high precision makes IIC attractive for repetitive SPM-probe positioning, in applications where precision is critical to performance. For example, IIC was use to reuce ynamics-couplingcause errors in the esire AFM probe-sample force 7, to enable AFM measurements of the fast, rate-epenent elastic moulus of soft polymers in Ref. 13, an for rapi measurements of the ahesion force on a silicon sample in Ref Research Directions in Feeforwar Control The use of feeforwar in three current research efforts: a SPM-base nanoscale combinatorial-science stuies, b imaging of large soft samples, an c image-base control for sensor-less SPM, are escribe below. Cantilever Probe Library of Samples Fig. 15 Schematic of nanoscale combinatorial science: screening a library of samples with an AFM probe Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /

14 5.1 Nanoscale Combinatorial-Science Stuies. In combinatorial-science stuies numerous combinations of samples are create an teste Integrating SPMs e.g., an AFM as illustrate in Fig. 15 with combinatorial science allows this paraigm to be applie for molecular- an atomic-level stuies. Increasing the SPM operating spee is necessary to increase the screening throughput of such libraries of samples. This requires both a high-spee SPM measurements of each sample, which require high-banwith vertical control; an b high-spee transitions of the SPM-probe from one sample to the next, which require high-banwith lateral control High-Spee Sample Property Measurements. Highbanwith vertical control is neee to measure rapily the properties of each sample in the library. For example, material properties such as ahesion, friction, or boning strength at the nanoscale can be obtaine from measurements of the interaction force between an AFM probe an the sample 137. Since material properties are rate an inentation time-profile epenent, the probe-sample force nees to be measure by following specifie sample-inentation curves. To avoi vibration-cause eviation from the specifie sample-inentation curve uring high-rate measurements, there is a nee for high-banwith, precision, vertical positioning. Recent works have shown that inversion-base iterative control can be use to increase the inentation rate uring sample property measurements. For example, IIC was use to enable AFM measurements of the fast, rate-epenent elastic moulus of soft polymers in Refs. 13, High-Banwith Lateral Sample-to-Sample Control. High-banwith lateral control is necessary to move rapily the SPM-probe from one sample to another, without resiual vibrations when measuring properties at each sample. Since the time spent in this transition from one sample to another cannot be use for active measurements, the goal is to minimize this transition time for increasing throughput. It is, however, important to reuce if not completely remove resiual positioning vibrations once the AFM probe has reache the next sampling position. This is an output transition problem, i.e., given an initial state x of the piezoscanner, the goal is to bring the output lateral eflection, say, y to a esire value y t f =ȳ an hol it constant afterwars, t t f. Previous works have shown than the integration of inversion an optimal control methos can lea to feeforwar inputs that enable rapi output transitions without resiual vibrations 64,139,14. Such methos can be aapte to esign fast sampleto-sample transitions in AFM-base combinatorial-science stuies. 5. Imaging Large Soft Samples in Liqui. Imaging of cellular features requires SPM scan imensions in the range 1 m 141. For example, the imaging of cell protrusions such as lamellipoia can require scan sizes in the orer of m see Refs. 14,143. However, current AFM systems are too slow to investigate nanoscale variations in shape an volume of cellular processes with relatively large features, especially when imaging soft cells without stiff cell walls such as human cells 144,145. Feeforwar methos play an important role in the evelopment of high-spee SPM for rapily imaging such soft samples Scan Frequency Versus Scan Size for Soft Samples. For the same scan size, achieve scan frequencies for soft samples in liqui 43,47,53 57 ten to be several orers of magnitue lower than the scan frequencies achieve in air as compare in Fig. 5. This reuction in scan frequency is because of the nee to maintain small probe-sample forces i.e., high precision vertical positioning to avoi amaging soft samples as oppose to har samples 39,146. For example, keeping the maximum force F max below.1 nn when imaging soft cells 145 using AFM cantilevers with low stiffness K s aroun.1 N/m requires positioning Scan Size Expaning Phase Fixe Phase (Data Collection) Time Shrinking Phase Fig. 16 Zoom approach 148,149 : the three scanning phases expaning zoom-out phase, fixe phase, an shrinking zoom-in phase to maintain small tip-sample forces uring the iteration process precision of F max /K s =1 nm or less. Variations in the sample s vertical features of more than 1 m which is typical in large soft cells translates to a positioning precision of less than.1% of the positioning range, which is challenging to achieve at high operating spees. 5.. Development of Feeforwar Vertical Control for Unknown Samples. Feeforwar control is not reaily applie to vertical positioning because the sample profile is not known a priori. For nonminimum-phase positioning systems, the noncausal, exact-inversion-base feeforwar schemes, even those implemente in the time omain, require some prior knowlege of the scan path. The stringent precision requirement in the vertical irection favors the use of iterative methos, which can achieve precision close to the sensor noise levels provie the iterations converge an the soft sample is not amage uring the iteration proceure The Problem of Large Forces During Iteration Proceure. The problem, with using iterative approaches, is to avoi large tip-sample forces as the iterative proceure converges in particular, uring the very first step in the iteration process. At the start of the iteration, the sample profile is unknown; therefore, it is ifficult to use the inversion metho to achieve SPM-probe positioning over the sample profile. This can lea to large tip-sample forces an sample amage at the very first iteration. One approach, to avoi such sample amage, is to use a slow scan for ientifying the sample profile at the start of the iteration process an then use the inversion proceure to fin the feeforwar input for faster scans. The problem is that this slow scan can take a very long time. Moreover, the sample profile coul change an the slow-scan image can be istorte by rift effects The Zoom Approach to IIC for Vertical Positioning. In SPM imaging applications, information from the previous scan line can be use to improve the positioning in the current scan line 147. The main iea is that the current scan profile is close to the previous scan profile an, therefore, the positioning precision can be improve by using information from the previous scan line to evelop the feeforwar input for the current scan line. The challenge is to avoi large forces in the first iteration step, which can be accomplishe by the zoom-out/zoom-in iterative approach 148,149. This iterative approach has three phases as shown in Fig. 16: a Start with a small scan area an expan graually, b fix the scan size at the esire value an image the sample, an c reuce the scan size to a small value. The zoom approach achieves small tip-sample forces because of the small scan size at the start of the iterations. Note that sample-profile variations with a small scan size are small in the first iteration step. Therefore, the resulting positioning errors an the tip-sample forces are also small. The rate at which the scan size is change uring the expansion an reuction phases is ajuste to ensure that the variations in the tip-sample force are small. The zoom approach was implemente to image relatively large soft samples in a liqui meium, in particular, to image soft hyrogel contact lenses in a saline solution using an AFM 148,149. The zoom approach enable imaging at higher scan frequencies at 3 Hz over a 1 1 m scan area. This repre / Vol. 131, NOVEMBER 9 Transactions of the ASME

15 V V k V k1 STM IIC I I k Image Comparison e k P P _ Fig. 17 Block iagram of image-base SPM control metho 9. At each iteration step, k, the SPM in this case a STM is use to acquire a low-spee an a high-spee image I an I k, respectively. These two images are compare with to etermine the positioning error e k, which is use by the IIC algorithm to etermine the input V k1 for the next iteration step to improve the SPM s positioning accuracy. Magnitue (b) Frequency (Hz) G stiffer piezoscanner 1 G *G* G () opt Fig. 19 Two approaches to increase positioning banwith of piezoscanner, where the system moel is G in Eq. 8. a Flattening the frequency response by using the optimal inverse as feeforwar G 1 opt in Sec with c-gain G as in Eq. 9. b Stiffening the system by increasing each resonantvibrational frequency z, p1, an p an the gain k of G in Eq. 8 by an orer of magnitue. sents about an orer of magnitue increase in the scan frequency with the use of the zoom-base approach, when compare with previous results see Fig. 5, for such large, soft samples in liqui. 5.3 Image-Base Control for Subnanometer-Scale Stuies. SPMs can achieve subnanometer-scale positioning, e.g., carbon atoms separate by.5 nm on a graphite surface can be image using a STM. However, the banwith at which STMs can achieve accurate subnanometer-scale positioning is limite by the inability to measure the position of the STM-probe s tip at high scan frequencies. For example, the resolution of external sensors uring high-frequency scanning is limite ue to sensor noise, which tens to increase with the scan frequency an temperature 15. Moreover, stanar external sensors cannot irectly measure the lateral position of a STM-probe s atomically sharp tip. Instea external sensors can only measure the position of a ifferent point on the STM scanner; the position of the STM-probe s tip can only be inferre from such measurements. In contrast, the vertical position of the STM-probe s tip can be measure using the extant tunneling current sensor. Thus, sensor eficiencies preclue the use of stanar feeback control techniques when highbanwith lateral positioning is require at subnanometer scales. To enable high-banwith subnanometer-scale lateral positioning, an image-base control methoology has been evelope that exploits the STM s extant imaging capabilities 9, Image-Base Control. To resolve the problems associate with using external sensors, an iterative image-base feeforwar control metho has been evelope to increase subnanometer-scale positioning banwith in Refs. 9,91. This approach, which uses the image-istortion to moel an compensate for ynamic effects, extens previously evelope methos that have use SPM-images to correct for positioning errors cause at low operating spees 151,15. A block iagram of this image-base metho is shown in Fig. 17. The main iea is to quantify the error in positioning the SPMprobe over the sample surface by using SPM-images of stanar calibration samples. As the calibration sample surface is fixe i.e., features o not vary, the istortion in the image can be use to quantify the positioning errors, moel the ynamics using the input-output ata 91, an correct the feeforwar input to the SPM 9 using inversion-base iterative control. This metho has been applie to control a STM uring subnanometer-scale imaging as seen in Fig. 18. A low-spee image, where accurate positioning has been achieve, is shown in Fig. 18 a, where the hexagonal lattice structure an normal atomic separation typical of the image HOPG surface are visible. As the scan frequency is increase, positioning accuracy is lost resulting in a istorte image as seen in Fig. 18 b. With the use of the image-base control, in Fig. 18 c, high-banwith nanopositioning is achieve note that the hexagonal lattice structure an normal atomic separation have been regaine. This achieve high-frequency scan at khz which was a ata-collection-harware limit in Ref. 91, with the image-base feeforwar approach, is approximately 6% of the lowest resonant-vibrational frequency of the system at 3 khz, which is more than six times the typical 1/1 limit in Fig Summary of Discussions The previous iscussions on feeforwar approaches are summarize in this section along with concluing remarks y (A) y (A) y (A) (a) x (A) -5-5 x (A) 5 (b) (c) x (A) Fig. 18 Image-base control has been applie to enable high-spee imaging of a HOPG sample in Refs. 9,91. a Accurate low-spee 1 Hz STM image of graphite. b High-spee khz STM image of the same graphite surface showing the istortion cause by positioning error. c High-spee khz STM image obtaine using image-base control 91. Journal of Dynamic Systems, Measurement, an Control NOVEMBER 9, Vol. 131 /