ARTICLE IN PRESS. Control Engineering Practice

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1 Control Engineering Practice 17 (29) Contents lists available at ScienceDirect Control Engineering Practice journal homepage: wwwelseviercom/locate/conengprac Set-point variation in learning schemes with applications to wafer scanners Marcel F Heertjes a,b,, René MJG van de Molengraft a a Department of Mechanical Engineering, Dynamics and Control Technology, Eindhoven University of Technology, Den Dolech 2, 56 MD Eindhoven, The Netherlands b ASML, Mechatronic Systems Development, De Run 651, 554 DR Veldhoven, The Netherlands article info Article history: Received 17 November 27 Accepted 18 August 28 Available online 5 November 28 Keywords: Finite impulse response modeling Multi-input multi-output feed-forward design Motion control systems (Nonlinear) Iterative learning control Wafer scanners abstract This paper presents a finite impulse response strategy to deal with set-point variation in learning schemes On the basis of converged learning forces obtained with learning control at a specific acceleration set-point profile, a finite impulse response mapping is derived to generalize the learned forces at a specific set-point toward arbitrary set-point profiles, thus relaxing the need for further learning The above strategy is applied to the motion control systems of a wafer scanner in a multi-input multi-output feed-forward setting, where a variety of set-point profiles is used Industrial potential is demonstrated via robustness to set-point variation and the improvements obtained in settling-time reduction & 28 Elsevier Ltd All rights reserved 1 Introduction In high-speed motion systems such as the reticle and the wafer stages of a wafer scanner (Van de Wal, Van Baars, Sperling, & Bosgra, 22) learning can significantly improve upon performance This is because of the repetitive nature of its scanning motion (Dijkstra & Bosgra, 22; Rotariu, Ellenbroek, & Steinbuch, 23; Rotariu, Dijkstra, & Steinbuch, 24) In learning, information of previous executions of a repeated motion is used to update a command (usually a force) needed to counteract the effect of such motion at future executions, see for example Bristow, Tharayil, and Alleyne (26), Cai, Freeman, Lewin, and Rogers (28), and Moore (1999) for learning algorithms and Gunnarsson and Norrlöf (21) and Tousain and Van der Meché (21) for learning designs Variation in the scanning motion, however, avoids the application of the resulting commands learned at a specific motion to be effective in achieving performance when applied during a different motion; see also Xu (1998), Xu and Tan (22), Dixon and Chen (23), Xu and Tan (23), and Rotariu, Ellenbroek, Van Baars, and Steinbuch (23) for a related problem statement For this purpose a finite impulse response mapping is used as proposed by Potsaid and Wen (24) The forces learned for a representative acceleration set-point are mapped onto a finite impulse response (FIR) model In the wafer scanning example, this is done prior to the process of wafer illumination whereas during this process the learned forces are replaced by generalized learned forces being the result of the finite impulse response model and the acceleration set-points at hand; this is different from a run-to-run control approach such as for example considered by Bode, Ko, and Edgar (24) which lacks in situ performance measurement (the wafer needs to be further processed) and in which all set-points are known In a general multi-input multi-output feed-forward setting, the advantages are twofold On one hand, learning during the process of wafer scanning is avoided This maintains the high standard of performance in terms of wafer throughput On the other hand, learning is based on a small sub-set of a generally large variation of wafer set-points This constitutes the efficiency of the method This paper has three contributions: (i) a generalized learning through FIR modeling for motion systems, (ii) a multi-input multioutput learning approach in case these motion systems are weakly coupled, and (iii) an application of learning in the field of industrial wafer scanners The paper is further organized as follows In Section 2, the wafer scanner application, which serves as an experimental benchmark, is discussed in terms of its relevant motion control sub-systems, in particular, the reticle and wafer stage Section 3 considers learning in the repetitive context of scanning motion Section 4 deals with FIR modeling for multiinput multi-output feed-forward design A performance assessment using examples from an industrial wafer stage is presented in Section 5 Section 6 summarizes the main findings of the paper Corresponding author at: Department of Mechanical Engineering, Dynamics and Control Technology, Eindhoven University of Technology, Den Dolech 2, 56 MD Eindhoven, The Netherlands Tel: ; fax: addresses: mfheertjes@tuenl, marcelheertjes@asmlcom (MF Heertjes), mjgvdmolengraft@tuenl (RMJG van de Molengraft) 2 Dynamics and control of wafer scanners During the lithographic manufacturing of integrated circuits (ICs) wafer scanners achieve performance by combining nano /$ - see front matter & 28 Elsevier Ltd All rights reserved doi:1116/jconengprac2884

2 346 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) scale resolution with optimized wafer throughput The wafer scanning process, see for example Groot-Wassink, Van de Wal, Scherer, and Bosgra (25) and Heertjes and Van de Wouw (26), can be described as follows Light from a laser passes from a reticle which contains an image, through a lens, which scales down the image, onto a wafer, see Fig 1 Both reticle and wafer are part of two separate sub-systems: the reticle stage and the wafer stage Each stage employs a dual stroke principle: a long stroke for large-range motion and a short stroke for accurate positioning The short-stroke modules can be modeled as floating masses which are controlled in six degrees-of-freedom on a single-input single-output basis A distinction is made between scanning directions and non-scanning directions The scanning directions, for example the x- and y-directions of the short-stroke wafer stage, are controlled on the basis of both feedback and feedforward The non-scanning directions, for example the x- and z-direction of the short-stroke reticle stage, are mainly controlled by feedback Each of the controlled short-stroke directions can be represented by the simplified block diagram representation of Fig 2 which can be considered in the more general context of motion control systems On the basis of an acceleration set-point a and resulting command r, a servo error signal e is constructed via e ¼ r y with y the actual position (or angle depending on the choice of axis) of the considered plant P, in this case a short-stroke stage The error signal e is fed into a feedback controller C fb that aims at disturbance rejection mainly induced by force disturbances f To obtain sufficient tracking accuracy, an SISO (and model-based) feed-forward controller C ff is added reticle stage lens The short-stroke (electro-)mechanics of both the reticle- and wafer stages in the individual directions are characterized by double integrator behavior along with the expression of higherorder dynamics In controlling such dynamics, the feedback controller C fb is chosen as a series connection of three filter blocks: a proportional-integrator-derivative (PID) filter, which aims at both disturbance rejection and robust stability, a secondorder low-pass filter to avoid high-frequency noise amplification, and several notch filters to deal with resonant behavior in the plant In the frequency-domain, the controlled electro-mechanics are characterized by the open-loop frequency response function O l ðjoþ ¼C fb ðjoþpðjoþ such as depicted in Fig 3 for the scanning y-direction of both a reticle and a wafer stage In Bode representation, this figure shows the characteristics derived from a closed-loop measurement (solid) along with the characteristics of a model (dashed) From this figure, it is concluded that robust stability is sufficiently guaranteed The reticle and wafer stage can be considered in the simplified MIMO context of Fig 4 Different from Fig 2, interaction between the feedback loops is modeled using cross-talk forces acting on the considered plants: f xy acting on P yy and f yx acting on P xx From stability point of view, these MIMO forces remain small enough to justify the SISO feedback/feed-forward design approach related to Fig 2 From performance standpoint, this is not the case Given the tight performance specifications this industry is faced with, it suffices to state that a zero settling control aim is not possible without at least including some MIMO characteristics in the feed-forward design This is the purpose of the proposed learning such as presented in the next sections Herein the databased MIMO feed-forward contributions are used atop the modelbased SISO feed-forwards The SISO feed-forwards, which strictly speaking become redundant, are kept as an initial estimate in achieving performance 3 Iterative learning control 25m wafer stage 45m Fig 1 Wafer scanner representation and its main components For controlled processes exhibiting repeated motion, iterative learning control provides a means to learn updated commands (or forces) from past servo information and apply these commands at future executions (or trials) of such motion, see also Chen and Hwang (25), Mishra, Coaplen, and Tomizuka (27), and Tayebi and Islam (26) The latter with the aim to improve upon servo performance The MIMO context in which iterative learning control is applied to the short-stroke stages of a wafer scanner is expressed by the simplified block diagram representation of Fig 5 Different from Fig 4, each axis contains a set of learning controllers C ilc 2 fc ilc;xx ; C ilc;xy ; C ilc;yx ; C ilc;yy g used to counteract the recurring error contributions e x and e y induced by the acceleration set-point profiles a x and a y For example, the x-axis contains a learning controller C ilc;xx which generates a force f ilc;xx This force is used to counteract the recurring part of the error response e x which is induced by the acceleration set-point a x ; note that this is the error C ff (s) a 1 r e C fb (s) - f ff f P (s) y Fig 2 Simplified block diagram representation of a motion control system

3 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) amplitude in db rs modelled ws modelled rs measured ws measured phase in degrees Fig 3 Bode representation of the measured open-loop frequency response functions of the short-stroke wafer (ws) and reticle (rs) stage dynamics in scanning direction along with the characteristics of two double integrator-based models C ff, xx f ff, xx C ilc, xx f ilc, xx a x 1 r x e x C fb, xx a y 1 r y e y C fb, yy f xy P xx P yy f yx y x y y C ilc, yx C ff, xx f ilc, yx f ff, xx C ff, yy f ff, yy Fig 4 Simplified block diagram representation of the MIMO controller structure such as encountered for both the short-stroke wafer and reticle stages a x 1 r x e x C fb, xx a y 1 r y e y C fb, yy f xy P xx P yy f yx y x y y response after application of the SISO model-based feed-forward controller C ff ;xx Additionally but applied in the y-axis C ilc;xy generates a force f ilc;xy used to counteract the recurring error response e y The latter being the result of cross-talk induced by the set-point a x The effectiveness of such a coupled system compensation stems from the assumption that the coupling is small enough to endanger SISO stability but large enough to improve upon MIMO performance Hence the wafer stage plant of about 225 kg, which is accelerated in the y-direction with 27:5ms 2, requires feed-forward forces near C ff ;yy 62 N Since f ilc;yy and f ilc;yx are typically in the order of 1 N, see Heertjes and Tso (27a), a sufficiently small coupling validates an essentially SISO-based stability approach However given a typical controller gain of Nm 1 such learning forces correspond to errors in the order of 5 nm which are large enough to allow for a significant MIMO performance enhancement The learned forces f ilc in Fig 5 comply with the schematics of Fig 6; see Heertjes and Tso (27a) for a detailed SISO C ff, yy C ilc, xy C ilc, yy f ff, yy f ilc, xy f ilc, yy Fig 5 Simplified block diagram representation of a simplified linear feedback connection of two coupled short-stroke axes having iterative learning control description That is, the error signals e related to a single execution of the acceleration set-point are stored in a buffer to form the error column eðkþ where k 2 N ers to the k-th execution (or trial) The buffer output is subjected to the nonlinear

4 348 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) weighting U, or UðeðkÞÞ ¼ diagðfðeðkþf1gþ; fðeðkþf2gþ; ; fðeðkþfngþþx, (1) with 8 >< fðxþ ¼ 1 d if jxj4d; jxj >: if jxjpd: All entries in eðkþ that are bounded in absolute value by a threshold level dx are assumed to be noise contributions and as such are excluded from learning; typically d is chosen equal to the L 1 -norm of the steady-state signals obtained after all transients have sufficiently damped out Though limited in fully capturing the noise characteristics, the choice for d along with the structure of (2) stems from both (nonlinear) stability and performance considerations such as given in Heertjes and Tso (27a, 27b) After this weighting, the error signals are subjected to a linear learning gain L which is given by L ¼ðS T p S p þ liþ 1 S T p, (3) with tuning parameter l4, see Ghosh and Paden (22) Basically, l is given the smallest possible value for which the learning process is stable, such that L closely resembles the e buffer C ilc buffer e (k) f ilc (k) f Φ ( ) ilc (k1) L z 1 I Fig 6 Block diagram representation of the iterative learning control f ilc (2) inverse process sensitivity S 1 p and a fast convergence is obtained S p 2 R nconn obs is given by 2 3 h 1 h 2 h 1 S p ¼, (4) h ncon h ncon 1 h ncon n obs þ1 where n con 2 N is the number of samples accessible for the learning force, the so-called controller window and n ob N is the number of samples used for error evaluation, named the observation window (Dijkstra & Bosgra, 22) S p has a Toeplitz structure where h 1 ; h 2 ; ; h n represent the Markov parameters h 1 represents the first error response sample e to a unitary force impulse f The Markov parameters are obtained from measurement and typically lect the average of 25 subsequent impulse response measurements as to reduce the effect of measurement noise, see also Ahn, Moore, and Chen (26) and Moore, Chen, and Bahl (25) for dealing with model uncertainty in this regard The learning filters C ilc;xx and C ilc;yx (see Fig 5) are both based on S p;xx, hence they relate to the error response e x induced by a unitary force impulse f x With S p, the updated learned forces f ilc ðk þ 1Þ are given by f ilc ðk þ 1Þ ¼f ilc ðkþþluðeðkþþeðkþ, (5) where f ilc ðkþ after buffering (see Fig 6) gives the learned forces f ilc For a detailed treatment of the stability and convergence properties of such a nonlinear learning controller, the reader is erred to Heertjes and Tso (27a, 27b) For a short-stroke reticle stage module, the need for MIMO learning in view of (settling) performance is illustrated in Fig 7 Given a representative acceleration profile a y, which in scaled form is depicted in the upper left part of the figure (dashed curve), the servo error signals in y- and z-directions show significant response to the variation included in this profile Note that this is the response encountered under SISO model-based 4 4 e y in nm e z in nm E y in nm 3 3 E z in nm 3 3 cpsd of E y in nm cpsd of E z in nm Fig 7 Time-series measurement of the error signals in y- and z-directions of a short stroke reticle stage (upper part) along with the non-recurring residuals (lower part) of 2 different scans

5 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) feed-forward conditions In the interval where performance is required, ie, the scanning interval of constant velocity, the responses e y and e z induce an undesired settling time needed for the error to become sufficiently small The reproducibility of the settling phenomenon is shown in the lower part of the figure By comparing 2 separately measured error traces obtained in subsequent trials it can be seen that during constant velocity (beyond t ¼ :35 s) the residual errors E roughly remain inside a noise bound of 3 nm for the y-axis (left part) and 3 nm for the z-axis (right part); E is defined as Efig ¼efig P n i¼1efig=n, with n ¼ 2 and i erring to an error trace realization A similar observation follows for the root-mean-square values such as obtained via cumulative power spectral density (cpsd) analysis, which is shown in the lower part of the figure In the MIMO context of Fig 5, the nonlinear learning control is applied to the reticle stage system with the aim of achieving zero settling times without transmitting too much noise through the learned forces The result of this is shown in Fig 8 For the nonlinear filter setting d y ¼ 3 nm and d z ¼ 3 nm, it can be seen that the servo errors after 1 trials (ilc) roughly remain below the threshold levels This gives a significant improvemenervo performance in comparison with the errors before learning () The amplification of noises through learning is kept small This is expressed by comparing 2 error traces With each trace being the result of a 1-trial learning applied under similar initial conditions, it can be seen that the noise level does not significantly differ from the noise level encountered before learning in Fig 7 Note that the idea of using ILC to compensate the recurrent part of the error but with a potential deterioration of the non-recurrent part is not restricted to the wafer scanner application, see, for example, Chen, Moore, Yu, and Zhang (28) in which hard disk drive servo performance is considered in terms of repeatable runouts (RRO) and non-repeatable run-outs (NRRO) In the wafer scanning process, the motivation for the application of learning techniques stems from the recurring acceleration set-point behavior during the process of wafer illumination At the same time, however, this process shows sufficient deviation in terms of set-points and servo errors to limit standard learning schemes from being effective This is illustrated in Fig 9 where a typical wafer scanning path during a job is depicted Starting from a fixed location outside the wafer (see the upper left part of the figure) an up-and-down scanning pattern in terms of third-order position set-points is commanded on a grid of equally distributed x- and y-locations along the wafer The resulting acceleration setpoints show much similarity which forms the basis of a recurring error response However variation occurs For example, at the crossings from a previous array of scans toward the next array (this is indicated by the dotted path) the acceleration set-points become different This mostly involves variation in the interval of y-position in mm wafer scanning path 3 mm wafer wafer stage top view x-position in mm Fig 9 Graphical representation of a wafer scanning path during a job e y in nm 4 ilc e z in nm E y in nm 3 3 E z in nm 3 3 cpsd of E y in nm cpsd of E z in nm Fig 8 Time-series measurement of the error signals in y- and z-directions of a short stroke reticle stage (upper part) before (gray) and after (black) learning (1 trials) along with the non-recurring residuals (lower part) of 2 different learned scans; l y ¼ 1 17, l z ¼ 1 15, d y ¼ 3 nm, d z ¼ 3 nm, n con ¼ 226, n obs ¼ 275

6 35 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) maximum acceleration where the time needed to reach the desired velocity varies, but also variation in the interval of maximum jerk, for example, when the desired velocity is reached prior to reaching the level of maximum acceleration All of this is related to the fact that the motion parameters such as maximum acceleration and jerk are generally fixed and chosen maximal as to reduce processing time As a result of set-point variation, however, the dynamics are excited differently inducing a different error response The previously learned forces based on earlier error behavior then become less effective or possibly even ineffective Since overall wafer performance is as good as the worst scan in a job such ineffectiveness is generally not acceptable and as such provides the motivation for including robustness against set-point variation in the learning process 4 FIR mapping In dealing with set-point variation, an FIR mapping is considered After convergence of the learning process, the ILC forces are mapped onto the corresponding acceleration set-point profiles using FIR modeling, see also Potsaid and Wen (24) For arbitrary scanning set-points, the resulting models are used to compute the generalized forces Essentially this reduces the learning strategy to a learned feed-forward design; see also Van der Meulen, Tousain, and Bosgra (28) and Baggen, Heertjes, and Kamidi (28) for learned feed-forward design based on optimization An MIMO learned feed-forward design is depicted in the simplified block diagram representation of Fig 1 Different from Fig 5 the FIR controllers C fir 2fC fir;xx ; C fir;xy ; C fir;yx ; C fir;yy g relate the acceleration set-points a x and a y with the generalized forces f fir 2ff fir;xx ; f fir;xy ; f fir;yx ; f fir;yy g For example, the acceleration setpoint a x serves as input to the FIR controller C fir;xx which generates a force f fir;xx used to counteract the error response e x induced by this set-point Additionally, C fir;xy is used to counteract the crosstalk effect of a x to the error response e y Note that C fir;xy is depicted in the upper part of the figure because a x serves as an input to the FIR mapping This is different from Fig 5 where C ilc;xy has e y as input and theore is located in the lower part of the figure The FIR controllers C fir are given in discrete time by f fir ðkþ ¼c 1 aðk þ n nc Þþþc n aðk n c þ 1Þ; k 2 Z, (6) with n nc the number of non-causal time samples, n c the number of causal time samples, and n ¼ n nc þ n c 2 N Herein it is important to note that the filter order n is often obtained by trial and error It is bound, however, by the following trade-off: choosing n too small limits the ability to describe the required feed-forward signals Choosing n too large involves the risk of over-fitting The coefficients c i with i 2f1; ; ng are obtained from a least-squares optimization That is, under the assumption that the learned (and converged) forces can be described by f ilc ¼ A ilc c, (7) with f ilc ¼½f ilc ðkþ f ilc ðk þ oþš T 2 R o1 a representative (training) set of ILC forces (oxn denoting the number of learned data points), the non-square matrix 2 a ilc ðk þ n nc Þ ::: 3 a ilc ðk n c þ 1Þ A ilc ¼ , (8) a ilc ðk þ n nc þ o 1Þ ::: a ilc ðk n c þ oþ with A ilc 2 R on, and c ¼½c 1 c n Š T 2 R n1, the FIR coefficients read c ¼ðA T ilc A ilcþ 1 A T ilc f ilc (9) a x 1 r x e x Note that although set-point variation occurs, during a wafer scanning job most of the scans are conducted under equal setpoint conditions This makes it intuitively clear to adopt these conditions as being most representative for training the set of ILC forces and obtaining the set of FIR coefficients Note moreover that the existence of a set of FIR coefficients c in (9) is based on the assumption that A T ilc A ilc is invertible The validity of this assumption and the related conditions on persistent excitation (PE) follow from the next result Theorem 1 Assume that A ilc in (8) is chosen such that a ilc ðk þ n nc Þa, oxn41, and A ilc does not contain columns having o equal entries, then A T ilc A ilc is invertible Proof See the Appendix The input output relation of the considered mapping has two important steady-state properties: first, a constant (acceleration) input gives a constant output and, second, an input with a constant slope gives an output with a constant slope For the class of third-order position set-points in which the acceleration profiles are composed of regions of constant output and regions of constant slope only, the first property follows from the fact that for a constant input aðmþ ¼¼aðm n þ 1Þ with mxn, the computed force f fir ðmþ in steady-state reads f fir ðmþ ¼c 1 aðmþþþc n aðm n þ 1Þ ¼ Xn i¼1 C fir, xx C fir, xy C ff, xx a y 1 r y e y C ff, yy C fir, yx C fir, yy f fir, xx f fir, xy f ff, xx C fb, xx C fb, yy f ff, yy f fir, yx f fir, yy Fig 1 Block diagram representation of a simplified feedback connection of a short-stroke x- and y-axes in a single direction having learned feed-forward control P xx c i aðmþ (1) f xy P yy f yx y x y y

7 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) The second property follows from re-writing (1), or f fir ðm þ 1Þ ¼f fir ðmþþ Xn i¼1 c i Da, (11) with Da ¼ aðm þ 1Þ aðmþ the fixed variation during one timesample For n ¼ n c ¼ 2 both properties can be distinguished from the FIR forces such as depicted in Fig 11 This figure also shows the corresponding ILC forces and acceleration profile; the latter being scaled with the sum of the filter coefficients, ie, P nc i¼1cðiþ In terms of FIR filter design, it can be seen that prior to the considered phases of constant slope, the ILC forces do not demonstrate the need for any non-causal filter contribution: the oscillations expressed by the ILC forces seem strictly noise related Theore, the mapping is studied in view of causal filter coefficients only, ie, n nc ¼ and n ¼ n c Note that the application of non-causal FIR filter coefficients can also be left to the least-squares optimization (by choosing n nc a) but with the possible effect of noise corruption by including non-relevant noise-, nonlinearity-, and non-set-point related contributions prior to the changes in the set-point profiles In the MIMO context of Figs 4, 5, and 1, Fig 12 shows the effect of: (i) no learning, (ii) an ILC such as discussed in the previous section, and (iii) a generalized learning control based on FIR modeling At a short-stroke wafer stage and in the absence of learning (), the effect of a single acceleration set-point in the x-axis (dashed curve) is clearly visible in terms of performance limiting error levels and settling times, see the upper part of the figure With learning (ilc), the error levels in both x and y-directions become significantly smaller In the middle part of the figure, it can be seen that the ILC forces f ilc;xx and f ilc;xy clearly relate to the set-point characteristics and show a good correspondence with the generalized FIR forces f fir;xx and f fir;xy (fir) This correspondence also applies to the error responses which can be seen both in time-domain (upper part) as well as in frequencydomain (lower part), the latter via cpsd analysis The ability to generalize the learned forces while maintaining servo performance is considered in Fig 13 In a cross validation experiment where two scanning velocities are considered: v ¼ 1:2ms 1 (left part) and v ¼ 2:4ms 1 (right part), the effect of learning is shown in terms of improved settling behavior At v ¼ 2:4ms 1, the learned forces (thick-gray) and corresponding set-point profile (dashed) are used to construct the FIR mapping f ilc, f fir in N 1 acc fir ilc 2 6 Fig 11 Time-series measurement of a learned force signal (thin) and the corresponding approximation (thick) along with the acceleration profile scaled with P nc i¼1cðiþ, nc ¼ 2 On the basis of this mapping, generalized FIR forces (thick-black) are applied to the coupled z-direction of a short-stroke reticle stage This is done at the set-point profile for which is learned but also for the set-point profile at v ¼ 1:2ms 1 for which in general is not learned Both in time-domain (upper part) as well as in frequency-domain representations (lower part), it can be seen that the generalizing properties of the FIR approach relate to a significant improvemenettling behavior, thereby showing a reasonable match with learning at each set-point separately In designing an FIR filter, the number of FIR coefficients n represents an important parameter in achieving servo performance To study its effect, different pairs of ILC forces and corresponding FIR models are compared, each pair related to a model based on a different number of FIR coefficients n ¼ n c 2f1; ; 55g The corresponding FIR forces are applied to a short-stroke wafer stage This is illustrated in Fig 14 where performance is assessed through the infinity norm of the nonscanning (coupled) error signal e rz Additionally, the effect is shown at a fixed number of coefficients n ¼ 3 by time-series measurement and cpsd analysis In a cross-validation context containing two scanning velocities v ¼ :6 and :3ms 1, it can be seen that beyond the large error reduction at n ¼ 5, only limited extra reduction is obtained by increasing the number of FIR coefficients The infinity norm is minimal at n ¼ 3, beyond which hardly any settling-induced behavior is found in the error signals The observation that a low-order feed-forward control design solves the performance problem to a large extent is widelyacknowledged in industrial motion control In this regard, the generalized FIR forces form no exception Apart from the number of FIR coefficients, the data interval used to construct the FIR mapping (this is indicated by n fir pn obs ) is an important design parameter This is because the linear mapping at hand demonstrates sensitivity to actuator/amplifier nonlinearity, 1 which is encountered when changing the direction of motion, the so-called direction dependency, but also when comparing the acceleration phase with the deceleration phase within a single motion Even when comparing the positive jerk (derivative of acceleration) phase with the negative jerk phase within a single acceleration (or deceleration) phase The effect is shown in Fig 15 at a reticle stage by considering two FIR controllers Each controller uses a different (but overlapping) interval of training data along the execution of the same acceleration set-point profile In the first interval, data is recorded from t ¼ 3:3 1 3 to 5:5 1 2 s(n fir ¼ 26) which covers the acceleration phase almost entirely In the second interval, data is recorded from t ¼ 3:2 1 2 to 5:5 1 2 s(n fir ¼ 118) which only contains data near the end of the acceleration phase From force time-series measurement, it becomes clear that the results related to interval 1 give a better approximation along the entire acceleration profile Since interval 2 only takes into account data near the end of the acceleration profile, the generalized forces at this part of the profile are very accurate accordingly From error time-series measurement this is the middle part of the figure it is shown that the forces based on interval 2 reduce the effect of settling behavior beyond the acceleration phase in a way similar to the ILC forces (indicated area) By evaluating the data (through the infinity norm) at the constant velocity phase, it can be seen that an optimum for n fir appears near the end of the acceleration phase (lower part) This is clear from the fact that earlier set-point excitation is largely excluded from the FIR mapping As a result, the FIR forces exactly match with the ILC forces beyond this phase The optimum is shown for the difference between the ILC and FIR 1 The amplifiers/actuators possess hysteresis as well as motor constant variation

8 352 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) e x in nm f ilc, xx, f fir, xx in N cpsd of e x in nm acc ilc fir e y in nm f ilc, xy, f fir, xy in N cpsd of e y in nm Fig 12 Time-series measurement of the servo error signals in scanning x-direction (left) and coupled y-direction (right) of a short stroke wafer stage before (gray), after learning (thin-black, 1 trials), and after the learned feed-forward control (thick-black); l ¼ 1 17, d ¼ 3 nm, n con ¼ 226, n obs ¼ 3, plus cumulative power spectral densities of the servo error signals e z in nm 4 v = 12 ms 1 acc ilc fir e z in nm 4 v = 24 ms 1 f ilc, yz, f fir, yz in N cpsd of e z in nm f ilc, yz, f fir, yz in N cpsd of e z in nm Fig 13 Time-series measurement of the servo error signals in the coupled z-direction of a short stroke reticle stage before (thin-black), after learning (thick-gray,1 trials), and after the learned feed-forward control (thick-black); l ¼ 1 15, d ¼ 1 nm, n con ¼ 22, n obs ¼ 325, plus cumulative power spectral densities of the servo error signals forces Df ¼ f ilc f fir (-symbols) but also for the servo errors e fir corresponding to the FIR forces (&-symbols), the latter being multiplied with the overall servo gain k p ¼ 2:3 1 7 Nm 1 Namely under the assumption that e ilc ¼, Df is related to e fir via the closed-loop process sensitivity function PðjoÞ e fir ðjoþ ¼ Df ðjoþ (12) 1 þ C fb ðjoþpðjoþ

9 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) e rz in μ rad 1 fir (6) fir (3) n c v = 6 ms 1 at n = 3 v = 3 ms 1 at n = 3 5 e rz in μ rad cpsd of e rz in μ rad acc fir e rz in μ rad cpsd of e rz in μ rad Fig 14 Time-series measurement of the error signal e rz evaluated (and cross-validated at two different scanning speeds) through the infinity norm and using FIR learning for a different number of FIR coefficients n 2f1; ; 55g in the non-scanning direction of a short-stroke wafer stage 1 interval 1 (n fir = 26) interval 2 (n fir = 118) 2 e y in nm f ilc, f fir in N ilc fir 4 6 in N interval 1 interval 2 4 Fig 15 Sensitivity of the finite impulse response mapping to expressions of nonlinear system behavior in the learned forces for the scanning y-direction of a short stroke reticle stage; l ¼ 1 17 m 2 N 2, d ¼ 5nm, n con ¼ 226, n obs ¼ 275, n ¼ n c ¼ 45 Below the bandwidth, it follows that ke fir ðjoþk kc fb ðjoþk 1 k Df ðjoþk For the considered PID design this can be simplified to ke fir ðjoþk kdf ðjoþk=k p From Fig 15, it is concluded that the presence of nonlinearity can severely limit the output of the mapping to resemble with the ILC forces and theore with the ILC error response

10 354 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) Performance assessment on a short-stroke wafer stage To demonstrate the potential of the learned feed-forward design in achieving robust performance, an industrial performance assessment is conducted on a short-stroke wafer stage In the analysis, robustness to set-point variation is tested by including different die sizes and scan velocities Performance is evaluated in terms of settling-time reduction In this regard, two industrial performance measures are considered: the moving average filter operation and the moving standard deviation filter operation (Heertjes & Van de Wouw, 26) The moving average filter operation expresses the level of position accuracy that can be obtained during the process of wafer scanning It has a strong relation with so-called scanning overlay (see also Bode et al, 24) and is defined as M a ðiþ ¼ 1 n win iþn win=2 1 X j¼i n win =2 eðjþ; 8i 2 Z, (13) with n win 2 N an application specific time frame This filter operation basically represents a low-pass filtering of the error signal e The moving standard deviation filter operation expresses the fading properties of the created image It is defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 iþn win=2 1 X M sd ðiþ ¼t ðeðjþ M a ðiþþ 2 ; 8i 2 Z, (14) n win j¼i n win =2 and as such has high-pass filter properties In the context of wafer scanner performance, Fig 16 shows the result of a robustness study to set-point variation At five different locations on the wafer (four corner points and the center point) eight similar scans at each location are executed and evaluated From these 4 measured time-series, the maximum absolute position error during scanning (denoted by kk 1 ) is depicted for both the short-stroke x- and y-axes, ie, a single value for each axis Nine of such values are obtained by repeating the experiment for three different scan velocities: :15; :3; :6ms 1, and three different die sizes: 8; 16; 32 mm, the latter lecting different position intervals centered about the considered wafer locations The experiments are executed without learning () and with a FIR controller (fir) thereby considering 72 different measured scans In terms of performance, Fig 16 shows that the infinity 6 6 fir e x in nm 3 e y in nm die size in mm 35 7 scan velocity in ms 1 4 die size in mm scan velocity in ms 1 Fig 16 Performance analysis from an unfiltered error viewpoint before (gray) and after (black) learned feed-forward control applied in scanning y-direction at a short stroke wafer stage at three different scan velocities: :15; :3; :6ms 1, and three different die sizes: 8; 16; 32 mm; l ¼ 1 17, d ¼ 3nm,n con ¼ 226, n obs ¼ 3, n ¼ n c ¼ 45 in nm 1 in nm 1 a (e x ) 5 a (e y ) 5 4 die size in mm scan velocity in ms die size in mm 35 7 scan velocity in ms 1 sd (e x ) in nm die size in mm 35 7 scan velocity in ms 1 in nm sd (e y ) 25 fir die size in mm 35 7 scan velocity in ms 1 Fig 17 Performance analysis from an M a- and M sd -filtered error viewpoints before (gray) and after (black) learned feed-forward control applied in scanning y-direction at a short stroke wafer stage at three different scan velocities: :15; :3; :6ms 1, and three different die sizes: 8; 16; 32 mm; l ¼ 1 17, d ¼ 3nm, n con ¼ 226, n obs ¼ 3, n ¼ n c ¼ 45

11 MF Heertjes, RMJG van de Molengraft / Control Engineering Practice 17 (29) norm of the error, which is mainly induced by the settling phenomenon, is significantly reduced under the given MIMO learned feed-forward; the case without learning demonstrates what is obtained under model-based SISO feed-forward control In terms of scan velocity and die size variation, Fig 16 shows limited sensitivity, hence sufficient robustness, along the considered parameter plane Performance in terms of M a and M sd is considered in Fig 17 It shows that small scan velocities give rise to small error levels Apart from the reduced amount of excitation related to such velocities, this is clear from the fact that n win scan velocity 1 Decreasing the scan velocity increases n win which in Eqs (13) and (14) has the effect of averaging out the settling phenomenon Figs 16 and 17 clearly demonstrate the ability of the generalized learned forces to achieve robust performance 6 Conclusions For industrial wafer scanners, learning control provides a powerful means to improve upon settling performance The generalization of the learned forces through FIR modeling adds the necessary robustness to set-point variation which would otherwise severely limit the benefits of learning This is because the wafer scanning industry lacks an exact repetition of set-points and demands a firsttime-right strategy during scanning In an MIMO learned feed-forward context, the occurrence of settling behavior, one of the major servo limitations on wafer throughput, is shown to effectively disappear Contrarily, the FIR approach appears sensitive in the presence of nonlinear system behavior This avoids posing a general design rule regarding the data interval needed to construct the FIR mapping Regarding the number of FIR coefficients, a design argument other than keeping its number small cannot be deduced from the considered crossvalidation experiments The FIR approach shows a good robustness to set-point variation whereas servo performance is not severely compromised when compared to the application of the ILC forces at the set-point for which is learned This is the outcome of a cross validation experiment At the same time, the FIR approach demonstrates its ability to achieve performance in an industrial environment In this regard, the approach shows potential in the broader context of industrial motion control systems Appendix The proof of Theorem 1 is given as follows By adopting the notation a ilc ðk þ n nc Þ¼a n, it follows from (8) that 2 a 2 3 n þþa2 nþo 1 % ::: % a n 1 a n þþa nþo 2 a nþo 1 a 2 A T ilc A n 1 þþa2 nþo 2 ::: % ilc ¼, a 1 a n þþa oa nþo 1 a 1 a n 1 þþa oa nþo 2 ::: a 2 1 þ ::: þ a2 o (15) where % indicates the symmetric counterparts Invertibility of (15) implies that all columns are independent Consider the first and the second column, or 2 a 2 3 n þþa2 nþo 1 a n 1 a n þþa nþo 2 a nþo 1 a n 1 a n þþa nþo 2 a nþo 1 a 2 n 1 þ ::: þ 6 a2 nþo (16) In view of symmetry, both columns are independent if the two diagonal terms differ In fact, this holds true for each consecutive pair of columns, ie, the second and the third, the third and the fourth, and so on If both diagonal terms are equal, or a 2 n þþa2 nþo 1 ¼ a2 n 1 þþa2 nþo 2 ) a2 n 1 ¼ a2 nþo 1, (17) independency results from the fact that the symmetric offdiagonal terms differ from the diagonal terms Namely assume all four terms in (16) are equal, or a 2 n þþa2 nþo 1 ¼ a n 1a n þþa nþo 2 a nþo 1 ¼ 1 2 a2 n þþ1 2 a2 nþo 1 þ 1 2 a2 n 1 þþ1 2 a2 nþo 2 Then (18) combined with (17) gives 1 2 ða n a n 1 Þ ða nþo 1 a nþo 2 Þ 2 (18) 1 2 a2 n 1 þ 1 2 a2 nþo 1 ¼ 1 2 ða n a n 1 Þ ða nþo 1 a nþo 2 Þ 2 ¼, (19) or a n a n 1 ¼¼a nþo 1 a nþo 2 ¼, which under the assumption that a ilc ðk þ n nc Þ¼a n a, reduces to a n 1 ¼ a n ¼¼ a nþo 1 a But this contradicts the earlier assumption that A ilc contains no columns with o equal entries Hence the symmetric off-diagonal terms differ from the diagonal terms rendering the 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