Projection-Based Iterative Learning Control for Wafer Scanner Systems

Transcription

1 388 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE 2009 Projection-Based Iterative Learning Control for Wafer Scanner Systems Sandian Mishra and Masayoshi Tomizuka Abstract In this aer, an iterative learning controller (ILC) that uses artial but most ertinent information in the error signal from revious cycles is emloyed for recision control of a wafer stage Tyically, ILC schemes use the error signal from the revious cycle for udating the control inut This error contains both reetitive and nonreetitive comonents The nonreetitive comonents of the error cause degradation of erformance of the ILC scheme Based on structural information about the lant and the disturbances, we can determine some basis functions along which the reetitive error is concentrated This information is extracted by rojecting the error signal onto the subsace sanned by these basis functions The rojected error signal is then used in the ILC udate law Stability and convergence conditions are resented for this rojection-based ILC udate law The roosed idea is motivated by recision control of a wafer stage For a constant velocity scan by the wafer stage, the major sources of reetitive error are found to be hase-mismatch and force rile These effects are mathematically modeled to obtain the subsace sanned by them The rojection-based ILC scheme using this subsace is then imlemented on a rototye one DOF stage and its erformance is comared to the standard ILC scheme that uses a frequency-domain filtering to remove nonreetitive comonents of the error Index Terms Iterative learning control, recision motion control, subsace rojection, wafer stage control NOMENCLATURE e k Lifted outut error vector at the kth cycle F Standard two-norm or induced two-norm on R N or R NxN r Lifted desired trajectory vector r k Lifted reference inut vector at the kth cycle w k (j) w(kn + j) y k Lifted lant outut vector at the kth cycle R(F ) Range sace of the matrix F ρ(f ) Sectral radius of the matrix F σ(f ) Largest singular value of the matrix F I INTRODUCTION Iterative learning control (ILC) has found widesread industrial alication in control of reetitive rocesses ILC is loosely based on the aradigm of human learning In a reetitive rocess, information from earlier iterations of the rocess can be used to imrove erformance in the current iteration The first rigorous formulation of ILC was develoed by Arimoto [1] and Uchiyama [2] Since then, ILC has been imlemented in several industrial rocesses because of its simlicity of design, analysis, and imlementation In articular, it has been successfully imlemented in comuter-numerical control tools [3], injection molding systems [4], microscale robotic deosition [5], among many Manuscrit received February 26, 2008; revised May 15, 2008 and July 25, 2008 First ublished March 27, 2009; current version ublished June 17, 2009 Recommended by Technical Editor Y Hori This work was suorted by UC Discovery Grant ELE10197 and by Nikon Research Center of America, Belmont, CA S Mishra is with the Deartment of Mechanical Science and Engineering, University of Illinois at Urbana-Chamaign, Urbana, IL USA ( sandian@illinoisedu) M Tomizuka is with the Deartment of Mechanical Engineering, University of California, Berkeley, CA USA ( tomizuka@meberkeleyedu) Color versions of one or more of the figures in this aer are available online at htt://ieeexloreieeeorg Digital Object Identifier /TMECH other examles ILC has also been used for inut shaing (learning inut shaing technique) for industrial robots [6] For recision control of hybrid steing motors, Cheng et al roosed a learning-based scheme for torque rile suression [7] A similar learning strategy for disturbance suression has been used in magnetic levitation systems [8] Alongwith alications, many ILC algorithms that guarantee better robustness, erformance, and faster rates of convergence have been develoed The P-tye, roortional derivative (PD)-tye ILC schemes develoed by Arimoto were designed rimarily based on stability constraints and were tuned for linear systems Similar schemes for continuous-time systems called anticiatory ILC have also been designed based on stability of the ILC loo and frequency-domain considerations [9] For guaranteeing robustness and erformance, design based on stability was found to be insufficient Bristow and Alleyne roosed frequency-domain-based ILC design using time-varying filters [10] for imroving erformance and robustness ILC design based on otimization of a cost function was introduced by Furuta and Yamakita [11] Frueh and Phan [12] used the term linear quadratic otimal learning control (LQL) to describe the ILC counterart of otimal linear quadratic regulator (LQR) roblems in standard control system design More recently, wavelet-transform-based ILC control strategies have been develoed [13] Significant research has also been done in design and analysis of monotonically convergent ILC algorithms [10] Photolithograhy is one of the central rocesses in semiconductor manufacturing Wafer scanners are otomechanical devices used for lithograhy As hotolithograhy technology becomes more sohisticated, better ositioning accuracy is necessary for the wafer and reticle stages in wafer scanners At the same time, in order to increase throughut, it is necessary to imrove the seed of ositioning Hence, there is a growing need to introduce advanced control techniques for recision ositioning of the wafer and reticle stages in hotolithograhy machines Considering the reetitive nature of wafer scanning, ILC has been used extensively for imroving trajectory tracking and reetitive disturbance rejection in wafer scanners [14], [15] In more recent develoments, ILC has also been used to generate imroved trajectories [16] for wafer stage ositioning The ILC scheme is imlemented as an add-on feature around the standard feedforward and feedback loos Though ILC is extremely effective in rejection of disturbances that reeat in every cycle of the oeration, its erformance is significantly degraded with introduction of disturbances that can vary from iteration to iteration Therefore, it is imortant to searate out the nonreetitive disturbances from the reetitive disturbances In a rocess with both reetitive and nonreetitive disturbances, one way to address this issue of efficient learning is by filtering out nonreetitive disturbances Filtering in the frequency domain has been used in many alications and design of Q-filters for learning has been the focus of much research [10] This aroach involves filtering out those frequency comonents from the learning signal at which nonreetitive disturbances dominate However, frequency-domain distinction between reetitive and nonreetitive disturbances is not always effective, esecially when there is significant temoral variation in the disturbance signal For examle, nonreetitive disturbances may dominate over reetitive disturbances in a certain time segment of the reetitive rocess, as oosed to being dominant in a certain frequency region In such scenarios, it is intuitive to incororate this temoral variation of the disturbance into the ILC scheme, ie, use a time-domain segmented ILC algorithm [17] While a lot of research effort has been directed at enhancing erformance based on imroved learning algorithm design, there remain some /$ IEEE

2 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE oen questions about utilization of data for most effective learning In other words, it is imortant to consider not only how to learn efficiently, but alsowhat information to learn In many learning systems, although the learning signal may have very high comlexity, most of the reeatable information that is needed for learning may be concentrated in only a few directions Some ILC schemes based on rojection of the control signal onto a subsace were develoed by Hamamoto and Sugie [18] This aroach is further develoed in [19] and the effect of nonreetitive disturbances is reduced by avoiding differentiation of the error signal The underlying assumtion in the aforementioned study is that nonreetitive disturbances are concentrated in higher frequency bands Basis functions are used in [20] to identify the lant to be controlled and the ILC udate law is derived based on this identified lant The basis functions are determined from the trajectory to be tracked These aroaches tend to ick basis functions for identification or udate of the ILC law based on the trajectory alone, rather than the lant to be controlled or the disturbances In order to reduce comutational effort in the learning rocess, Ye and Wang have roosed discrete cosine transform-based rojection ILC algorithms in [21] This aer is based on the use of basis functions (or vector directions) for ILC based on a hysical model of reetitive disturbances as well as the trajectory to be tracked We roose an ILC algorithm that uses only artial error information from the revious iteration This artial information is obtained by rojection of the entire error information vector onto a smaller subsace sanned by a set of basis vectors The basis vector set should be chosen such that most of the reeatable error information is catured in the rojected error, while avoiding roagation of nonreetitive error Therefore, the choice of the basis vector set is motivated by the hysical system itself alongwith the task it executes The aer is organized as follows Section III resents the formulation of the rojection ILC roblem alongwith stability and convergence conditions for the rojection ILC scheme Section IV rovides exerimental results on a 1-DOF rototye wafer stage system Finally, conclusions and emerging ideas in rojection ILC are resented II PROJECTION ILC Let us consider a stable discrete-time linear time-invariant singleinut single-outut system, denoted by G(z 1 ), with a known delay n d This system executes a reetitive rocess with eriod of N samles We want the outut of the system to track a trajectory r(j), wherej ranges from 0 to N 1 This is reeated several times, with the system coming back to rest condition at the end of each iteration of the cycle, and starting at rest condition at the beginning of each iteration The outut of the lant for each iteration is denoted by y k (j), wherej ranges from 0 to N 1,andk denotes the iteration number Therefore, we have the following relationshi: y k (j) =G(z 1 )(r k (j)) (1) e k (j) =r(j) y k (j) (2) where r k (j) is the inut to the lant and e k (j) is the error from the desired trajectory r(j) Standard ILC roblem formulation: We can rewrite the revious equations in lifted form by stacking all the signals into N 1 vectors [15] Assuming zero initial conditions, we get the following lifted formulation of the ILC system: y k = T (r k + d) lifted lant equations (3) e k = r y k lifted error equations (4) where T is a Toelitz matrix comosed of the Markov arameters of the linear system G(z 1 ) g(0) 0 0 g(1) g(0) 0 T = (5) 0 g(n 1) g(n 2) g(0) where g(i) is the ith term of the imulse resonse In the general ILC design case, we have r k +1 = r k + Le k lifted learning equation (6) e k +1 =(I N TL) e k error evolution equation (7) The ILC scheme is stable if ρ (I N TL) < 1 Further, the ILC scheme is monotonically stable in the sense of the two-norm in R N if σ (I N TL) < 1 This is a desirable roerty to have if we wish to avoid oor learning transients Projection-based ILC roblem formulation: Consider a set of m orthonormal basis vectors b 1,b 2,,b m in R N Construct B =[b 1 b 2 b m ] R N m V B = R(B) (8) In rojection ILC, the rojection of the error e k onto the subsace V B (denoted by ē k ) is used in the learning udate law The learning system resulting from this is shown as r k +1 = r k + Lē k lifted learning equation (9) ē k = BB e k =Πe k rojection equation (10) e k +1 =(I N TLΠ) e k error evolution equation (11) In the following section, we will develo stability conditions and erformance bounds for the rojection ILC roblem A Stability Analysis For stability of the rojection ILC system, we roose the following lemma Claim 1: The ILC system described by (90 (11) is marginally stable [22] if ρ (I m B TLB) < 1 Proof: The roof is straightforward; therefore, it has been omitted Claim 2: The rojection ILC scheme described by (9) (11) is stable if the original ILC scheme (6) and (7) is monotonically stable (ie, σ (I N TL)=γ<1) ( Further, the error converges to the steady state error e ss = B B +( B TLB)(B TLB) 1 B ) e 0,whereB is an orthonormal set of vectors b m +1,b m +2,,b N, which, together with the vectors in B, constitute an orthonormal basis for R N Proof: Define B =[b m +1 b m +2 b N ] R N (N m ) e k =[B B ] [ zb z B The error evolution equation (11) then becomes [ ] [ zb Im B TLB 0 = B TLB z B k +1 ] k I N m z B,k+1 =(I m B TLB) z B,k and z B,k+1 = z B,k B TLBz B,k ][ zb z B ] k

3 390 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE 2009 Since σ (I N TL)=γ<1, lim k z B,k =0 Further, k 1 z B,k = z B,0 + ( B TLB)(I m B TLB) j z B,0 j =0 lim k z B,k = z B,0 +( B TLB)(B TLB) 1 z B,0 Therefore, ( e ss = B B +( B TLB)(B TLB) 1 B ) e 0 (12) qed The revious result relates the stability of the rojection ILC scheme to the monotonic stability of the original standard ILC scheme The choice of basis vectors is arbitrary as long as they are orthonormal Further, the residual steady-state error e ss has comonents only along the directions in B Therefore, we eliminate all reeatable error concentrated in the subsace sanned by B It can easily shown that the rojection of the error onto R(B) decreases monotonically if σ (I N TL)=γ<1 B Determining the Projection Subsaces The result in Claim 2 gives us a condition for designing the learning matrix L; however, we must also address the issue of choice of the basis vectors B In order to illustrate the revious oint, we analyze the effect of reetitive and nonreetitive comonents of the error on learning The error vector e k consists of the reetitive comonent e r,k and the nonreetitive comonent e n,k The error evolution equation for the rojection ILC scheme (with the learning filter L) is e r,k+1 =(I TLBB ) e r,k TLBB e n,k e k +1 = e r,k+1 + e n,k+1 In order to choose the basis function set B, we consider two asects First, TLBB e n,k should ( be small to avoid the effect of nonreetitive error Second, B B +( B TLB)(B TLB) 1 B ) e r,0 (the steady-state error) must also be small Therefore, we should ick those directions where the reetitive comonents of the error dominate the nonreetitive comonents In other words, the basis vectors B must be chosen such that they cature most of the reetitive information Remark 1: In many ILC alications, the error signal is often filtered by a low-ass filter This is consistent with the idea that reetitive comonents of the error dominate nonreetitive comonents at lower frequencies, while at high frequencies, nonreetitive disturbances dominate Remark 2: The authors would like to emhasize that the idea of rojection-based ILC schemes is not new [19], [20], [23] Basis functions are used in [20] to identify the lant and track the desired trajectory The basis functions are designed based on the reference trajectory Outut differentiation is avoided in [19] by using basis functions that are derivatives of the reference trajectory By avoiding outut differentiation, noise effects are suressed A higher order ILC algorithm is roosed in [19] to imrove noise tolerance An identification method based on an iterative learning control (ILC) for a class of linear multiinut multi-outut (MIMO) continuous-time systems with unknown but fixed inut disturbances is resented in [24] As in standard control design, in order to extract maximum erformance, reference tracking as well as disturbance rejection must be achieved simultaneously, and therefore, structural information regarding both the reference and the disturbance must be incororated into design of the basis functions A similar aroach to this aer using a disturbance model is used in [25] Fig 1 Schematic of stage/countermass system for learning This method requires ole assignment for a system matrix, the size of which deends on the number of basis functions used to reresent disturbances Thus, when this number is large, the method may become difficult to aly, for examle, a collection of signature functions corresonding to each samle oint of the cycle This aer stands aart from revious work in that it rooses an algorithm that enables the control engineer to use information regarding the exected reetitive disturbances in addition to the reference trajectory without the need for exlicit ole lacement for large matrices By exloiting the orthogonality of the basis functions and the monotonicity of the original learning scheme, the ole lacement roblem has been avoided III PROJECTION ILC IMPLEMENTATION FOR WAFER STAGE CONTROL A Exerimental Setu The single DOF wafer stage setu, shown in Fig 1, includes a stage and countermass system, both driven by linear motors The stage is mounted on air bearings and the countermass is guided by roller bearings Stage osition is measured by a laser interferometer and the countermass osition is measured using a linear encoder Though not shown, ower cables and neumatic tubing are among the otential sources of disturbances to the stage The stage can be modeled as a simle mass damer system as P (s) = s 2 +72s (13) The reference trajectory is designed as shown in Fig 2 This trajectory relicates the movement of one of the axes of a wafer stage The goal is to achieve constant velocity as soon as ossible to within a certain accuracy so that scanning can be erformed This imitates one scan, which is then reeated to roduce the same attern on multile ICs The overall control system is shown in Fig 3 The feedback PID controller C(z 1 ) was designed keeing in mind the bandwidth (120 Hz) and sensitivity constraints The samling time of the controller was set to 400 µs Inertial feedforward (ff(j) in Fig 3) was also used to imrove the transient resonse of the overall system ( ) C(z 1 T s 1 z 1 ) = (14) 1 z 1 There are two distinct hases of the trajectory: nonzero acceleration and zero acceleration During the acceleration hase, we exect the erroneous feedforward signal arising from modeling mismatch to T s

4 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE cause the dominant comonent of the following error However, during the constant velocity hase, we exect force rile, cable forces and vibrations to be the major sources of error Here, it is imortant to distinguish between the two tyes of disturbances resent in each hase Cable forces and vibration (and other nonreetitive disturbances such as measurement noise) cannot be learnt over cycles, since they are not constant over cycles On the other hand, the feedforward signal can be corrected and the force rile eliminated using ILC, since they do remain constant over cycles Fig 4 shows a lot of following error versus time for the first iteration It is interesting to note that during the acceleration hase, the major comonent of the following error is caused by modeling mismatch During the constant velocity hase, force rile is the major source of disturbance and results in the dominant comonent of the tracking error We will use this information about the sources of reetitive error to construct our basis vectors in B Fig 2 Fig 3 Reference trajectory Block diagram of overall control system B Construction of the Basis Force rile can be modeled simly as a sum of sine and cosine functions of osition and inut current [26] Equation 15 shows this model The variable x is the osition, is the electrical itch of the motor, u is the motor current inut, and F rile is the force rile F rile = u l ( ) 2πx 2πx a n sin n + b n cos n (15) n =0 During the constant velocity hase, the effect of this force rile can be aroximately modeled as (with v d as the desired constant velocity) l ( ) e rile (j) = α n sin nj + β n cos nj n =0 (16) Note that α n,β n are unknown rior to the exeriment run; however, they are fixed from run to run On the other hand, during the acceleration and deceleration hases, the effect of the hase mismatch becomes the major source of reetitive error; therefore, we will use the entire error signal during this art of the cycle Therefore, we first construct B r as B r =[b rs1 b rc1 b rs2 b rc2 b rcn ] sin n cos n sin (2n) cos (2n) b rsn = sin (3n) b rcn = cos (3n) 2πvd T s 2πvd T s sin (Nn) cos (Nn) This set of basis vectors describes the effect of the force rile Next, we will construct another set of basis vectors for the acceleration and deceleration hases This is described by B a given as B a =[q 1 q 2 q l ] (17) Fig 4 First cycle following error with no ILC q l (j) =0, if j l, j [0,N 1], else q l (j) =1 (18) We include those q l in B a for which l is art of the acceleration or deceleration hase Now we have a set of vectors [ B r B a ] that are not orthonormal Using a Gram Schmidt decomosition on the revious set of vectors, we obtain the set of orthonormal vectors B Considering the secific scan trajectory, we have, in all, 2016 basis functions C Results In this section, we resent an exerimental comarison of the standard filter-based learning algorithm and the roosed rojection-based learning algorithm In order to imrove the tracking erformance of the

5 392 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE 2009 Fig 5 Following error for iterations three, six, and nine with standard ILC using a 120-Hz filter Fig 7 Detail of actual and aroximated following error for the constant velocity scan hase in the acceleration hase, the following error is not aroximated Fig 6 Detail of following error for iterations three, six, and nine during constant velocity scan hase with standard ILC using a 120-Hz filter reetitive scanning rocess, an ILC loo was imlemented, as shown in Fig 3 The learning filter L(z 1 ) was designed as Fig 8 Following error for the entire cycle using the rojection-based ILC Scheme At the beginning and at the end of the constant scan hase, there is degradation of erformance During the acceleration and deceleration hases, the erformance of the rojection ILC scheme matches with that of the standard ILC scheme L(z 1 )=09z 2 F (z)f (z 1 ) (19) where F (z 1 ) is a first-order discrete-time filter with a desired cutoff frequency at 120 Hz, obtained from the standard filter design toolbox in MATLAB F (z) is obtained by relacing every z 1 in F (z 1 ) by z F (z)f (z 1 ) is acaudal, but has zero hase In order to guarantee the monotonicity condition necessary to imlement a stable rojection-based ILC scheme, we verify that σ (I TL)=093 < 1 (since 1 09 z 2 F (z)f (z 1 )G(z 1 ) < 1), where L was the lifted form of L(z 1 ) and T was the lifted form the of closed-loo transfer function G(z 1 ) The conventional (ie, nonrojection tye) learning law is, therefore, r k +1 = r k + Le k (20) The trajectory following error after three, six, and nine cycles is shown in Fig 5 We observed that following error convergence was raid in the acceleration hase However, in the constant velocity scan hase, tracking was significantly degraded Fig 6 shows a section of the constant velocity scan hase illustrating this This erformance degradation can be exlained by the fact that the learning system icked u significant nonreetitive disturbances like vibration during the constant velocity hase The eak scanning error was almost 700 nm In a second set of exeriments, a rojection of the lifted error vector onto the subsace sanned by B (derived in the revious section) was used in the learning scheme The learning filter L remained unchanged from the revious exeriment Therefore, r k +1 = r k + LBB e k (21) During the constant velocity scan, a very good aroximation of the error (BB e k ) could be obtained rojecting the error signal (e k ) onto the subsace sanned by B, as illustrated in Fig 7 A lot of the error for iterations three, six, and nine is shown in Fig 8 We observe that the following error during the acceleration and deceleration hases is comarable to the case when the entire segment was used in learning Performance was somewhat degraded at the beginning and end of the scan hase This can be attributed to the fact that all frequencies could no longer be learnt during the constant velocity scan hase However, during the constant velocity scan hase, significantly better tracking was obtained since the nonreetitive disturbances were not learned, as shown in Fig 9 The eak scanning error was reduced to under 150 nm by avoiding learning of the nonreetitive disturbances

6 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL 14, NO 3, JUNE Fig 9 Following error for the constant velocity scan hase using a rojection aroximation of the learning signal during the constant velocity scan hase, the tracking error is under 150 nm IV CONCLUSION Wafer scanning requires ultrahigh-recision ositioning caabilities In addition to smart design techniques, advanced control schemes are imortant for achieving the stringent erformance standards Iterative learning control imroves erformance of the wafer stage from scan to scan, since the rocess is reetitive The erformance of ILC schemes is significantly degraded in the resence of nonreetitive error In this aer, a method to reduce the effect of nonreetitive error on learning is roosed The reetitive error is extracted from the nonreetitive comonent of the error by rojection of the error signal onto a subsace The basis for this subsace is obtained from the hysical model of the lant and nonlinearities/disturbances As in standard control design, in order to extract maximum erformance, reference tracking as well as disturbance rejection must be achieved simultaneously Therefore, structural information regarding both the reference and the disturbance must be incororated into design of the basis functions In the case of wafer stage scanning, force riles and model mismatch were identified as the major sources of reetitive error, while noise, cable forces, and vibration were the major sources of nonreetitive error Considering this, the rojection subsace was obtained and the ILC udate law used only the comonent of the error signal in this subsace By using this technique, the effect of nonreetitive disturbances on the learning udate was reduced and much better tracking erformance was obtained, articularly during the constant velocity scan hase This general idea maybe used in other reetitive rocesses where the structure of reetitive (and/or nonreetitive) disturbances is known ariori REFERENCES [1] S Arimoto, S 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