Nonlinear Q-filter in the learning of nano-positioning motion
systems
Citation for published version (APA):
Heertjes, M., Rampadarath, R., & Waiboer, R. (2009). Nonlinear Q-filter in the learning of nano-positioning motion systems. In Proceedings of the European Control Conference 2009 (ECC'09), 23-26 August 2009, Budapest, Hungary (pp. 1523-1528). [MoC12.4] Institute of Electrical and Electronics Engineers.
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Nonlinear Q-filter in the Learning of Nano-Positioning Motion Systems
Marcel Heertjes, Randjanie Rampadarath, and Rob Waiboer
Abstract— To avoid an increased noise response under high-gain learning, a Q-filter with varying cut-off frequency is proposed. The Q-filter design is of particular interest in the wafer scanning industry where nano-position accuracy should be achieved under high-speed repetitive motion. In a lifted iterative learning control (ILC) setting, the nonlinear Q-filter is given state-dependent low-pass filter characteristics. Being induced by sufficiently large servo error signals, the Q-filter acts as a low-pass filter with sufficiently large cut-off frequency as to allow for a large learning gain, hence fast error convergence. For small error signals, i.e., the signal levels typically associated with noise, the Q-filter acts as a low-pass filter with a signifi-cantly reduced cut-off frequency. As a result, the amplification of noises through high-gain learning is kept limited. For a long-stroke wafer stage module of a wafer scanner, the effectiveness of the learning approach is assessed through experiment.
Index Terms— iterative learning control, Lyapunov stability, nonlinear control, Q-filter design, wafer scanner
I. INTRODUCTION
In high-speed (and nano-scale) positioning systems such as the stages encountered in the wafer scanning industry, high-speed motion is combined with nano-scale tracking precision. In terms of achieving servo performance, the combination of both speed and accuracy puts heavy demands on the control systems and design. To improve upon the wafer scanning performance in terms of wafer throughput, the need for more advanced control techniques such as learning control then becomes evident. Learning may result in the elimination (or at least reduction) of the recurring settling behavior prior to scanning which results from the repetitive nature of the scanning process and which currently limits a further reduction of the scanning time.
In motion systems, iterative learning control is a well-known technique to improve upon the error response under repetitive motion, see for example [4], [6], [12], [15] for tutorial works and [2], [5], [9], [11], [13], [14], [16], [18] for some recent developments and innovations in this field. Both in frequency (conventional ILC) as well as in time-domain (lifted ILC) representation, learning schemes have demon-strated to be effective by covering a wide range of industrial applications. But essentially offering solutions in the context of linear discrete-time feedback systems, these schemes are also bound to linear design limitations. For example, the
M.F.Heertjes is with Eindhoven University of Technology, Department of Mechanical Engineering, The Netherlands, email: m.f.heertjes@tue.nl and with ASML, Mechatronic Systems Development, The Netherlands, email: marcel.heertjes@asml.com
B.R.Rampadarath is with Eindhoven University of Technology, Department of Mechanical Engineering, The Netherlands, email: b.r.rampadarath@student.tue.nl
R.Waiboer is with ASML, Mechatronic Systems Development, The Netherlands, email: rob.waiboer@asml.com
error convergence over subsequent trials is coupled directly to the degree of learning expressed by the learning gains: large learning gains induce fast error convergence. But large learning gains are associated with the amplification of noises through learning. As a result most learning schemes balance error convergence with output noise amplification. In so doing, these schemes generally include some form of low-pass filtering (so-called Q-filtering) to prevent high-frequency signals from entering the learning feedback loop. Having significantly improved upon the robust stability prop-erties of the ILC and thereby allowing for larger learning gains and subsequent fast error convergence, the potential of the learning scheme to effectively deal with high-frequency repetitive error behavior is often compromised.
To balance more effectively the trade-off between error convergence and output noise amplification, this paper con-siders a nonlinear Q-filter. For error signals outside a pre-defined saturation band, the (nominal) Q-filter acts as a low-pass filter with a sufficiently large cut-off frequency. Inside the saturation band, however, the Q-filter primarily acts as a low-pass filter with an adapted (and much smaller) cut-off frequency. On one hand large error signals are subjected to high-gain learning, the latter inducing fast error convergence. On the other hand small error signals typically associated with noises are heavily low-pass filtered as to avoid the presence and amplification of these noises through learning. For a multi-variable long-stroke wafer stage, the effec-tiveness of the nonlinear Q-filter is demonstrated in ex-periment. The associated performance in terms of reduced settling behavior shows significant improvement. Bounded-input bounded-output (BIBO) stability of the nonlinear ILC follows from Lyapunov analysis. In fact, arbitrary (but finite) initial conditions are shown to converge to a positively invariant set about zero.
The paper is organized as follows. In Section II the dynamics and nominal control of a long-stroke wafer stage are considered. This provides the key motivation for the application of learning control as a means to improve upon the settling behavior. In the context of lifted ILC, Section III considers the proposed nonlinear Q-filter as a means to reduce the amplification of noise through learning. Lyapunov stability of the nonlinear ILC is addressed in Section IV whereas a performance assessment of the nonlinear ILC applied to the long-stroke wafer stage is presented in Section V. This paper is concluded in Section VI.
II. DYNAMICS ANDCONTROL OF ALONG-STROKE
WAFERSTAGE
To achieve nano-scale position accuracy with high produc-tion speed, the scanning stages of a wafer scanner exploit a dual-stroke principle: a long-stroke for course positioning and a short-stroke for fine positioning. An example of a long-stroke wafer stage is shown in Figure 1. Its
multi- = XXXXz
y
x
Fig. 1. Long-stroke wafer stage motion system.
body dynamics consist of a y-beam and an x-carrier and is controlled on the basis of a single-input single-output (SISO) control design. Herein the x-carrier supports the so-called mirror block which represents the short-stroke wafer stage.
A simplified MIMO representation of the SISO-based control design is given in Figure 2. In the horizontal plane,
r m -6 ? m-e s fff Cfb -? Cff - m- m-? f P y -6 + + -+ + +
Fig. 2. Long-stroke wafer stage MIMO control scheme.
e(t) = [ex(t) ey(t) erz(t)]T represents the servo error col-umn, which results from subtracting the plant output column
y(t) = [yx(t) yy(t) yrz(t)]T from the synchronized set-point column r(t) = [rx(t) ry(t) rrz(t)]T. The error is summed with a set-point update column s = [sx(t) sy(t) srz(t)]T resulting from an iterative learning control step (this is the topic of Section III) before being subjected to a diagonal controllerCfb with main entriesCfb,x,Cfb,y, andCfb,rz. The output of the feedback controller is summed with the output of a feedforward controller, i.e., fff(t) = [fff,x(t) fff,y(t) 0]T. The feedforward controller Cff is a diagonal (inertia-based)
controller with only two main entries: Cff,x and Cff,y. This is because there is no set-point in rz-direction giving rrz= Cff,rz= fff,rz= 0. Along with the feedback and feedforward controller contributions, plant disturbances are added through the input disturbance column f(t) = [fx(t) fy(t) frz(t)]T before being subjected to the MIMO plantP.
The feedback controllers are given by a series connec-tion of a proporconnec-tional-integrator-derivative (PID) control, a second-order low-pass filter, and two second-order notch filters. The simplified feedforward controllers are given by an inertia matrix multiplied with the second-order derivative of the set-point r.
Example 2.1: For the x-direction, explicit expressions of the feedback filters are given in Laplace representation by
Cfb,x(s) = Fpid,x(s)Flp,x(s)Fn,1,x(s)Fn,2,x(s), (1) with Fpid,x(s) = kp,x 1 + ωi,x s 1 + s ωd,x , (2) with kp,x = 4 105 Nm−1 the controller gain, ωi,x =
2π rads−1 the integrator breakpoint, and ωd,x= 21π rads−1
the differentiator breakpoint. The second-order low-pass fil-ter is given by Flp,x(s) = ωlp,x2 s2+ 2β lp,xωlp,xs+ ωlp,x2 , (3)
with ωlp,x= 160π rads−1 its cut-off frequency and βlp,x=
0.5 its dimensionless damping coefficient. The notch filters Fn,i,xwith i∈ {1, 2} are given by
Fn,i,x(s) =
ωp,i,x2
ωz,i,x2
s2+ 2β
z,i,xωz,i,xs+ ωz,i,x2
s2+ 2β
p,i,xωp,i,xs+ ωp,i,x2
, (4) with ωz,1,x = ωp,1,x = 224π rads−1 the zero and pole frequencies, respectively, of the first notch filter Fn,1,x and
βp,1,x = 10βz,1,x = 0.4 the corresponding dimensionless damping coefficients.The second notch filter’s parameters are given by: ωz,2,x = 90π rads−1, ωp,2,x = 110π rads−1,
βz,2,x= 0.1, and βp,2,x= 0.4.
Robust stability of a MIMO controlled multi-body model (multi-body because of the moving y-beam along with the moving x-carrier, see Figure 1) of plant P is sufficiently guaranteed given the MIMO Nyquist representation in Figure 3. By plotting the determinant of the 3 × 3 open-loop frequency response matrixCfb(jω)P(jω) at each considered frequency entry it can be seen that the three curves (asso-ciated with the horizontal plane coordinates x, y, and rz) demonstrate sufficient stability margins.
Performance of the MIMO controlled multi-body plantP both in terms of measurements and simulations is shown in time-domain representation in Figure 4 by plotting the error column e(t) induced by a representative set of
set-point profiles r(t). The upper part of the figure shows that
transient behavior (indicated area) dominates the response beyond the dictated set-point profile (dotted curves). The lower part of the figure shows the differences between
−1.5 −1 0 0.5 −1.5 0 0.5 real part im ag in ar y p ar t
Fig. 3. MIMO open-loop frequency response functions in Nyquist representation in the horizontal plane: x, y, and rz.
0 0.5 −400 0 400 0 0.5 −400 0 400 time in s ex ,ey ,er z in µ m er ro r in in µ m
model (dashed) versus measurement (solid)
Fig. 4. Time-series measurement and simulation of the error responses in the horizontal plane: ex, ey, and erz.
the simulated and measured errors. Though non-zero, these differences are assumed to be small enough to validate the model assumptions.
Since performances in the long-stroke wafer stage are expressed by the settling times needed for the errors to become sufficiently small, improved performance involves a set-point shaping s(t) as to reduce these settling times;
see also Figure 2. Because of the repetitive nature of the stage motion, it is natural to derive such shaping through the application of a learning scheme. The nonlinear Q-filter characteristics used in such a learning scheme will appear key to the stability and performances of the learning in the presence of uncertainties and disturbances.
III. ITERATIVE LEARNING CONTROL WITH NONLINEAR
Q-FILTERING
In literature [17], iterative learning control (ILC) appears to be an effective (if not the most effective) tool for dealing with recurring error behavior. Especially in time-domain rep-resentation, its lifted system description allows for
straight-forward implementation and analysis and easy modification toward nonlinear operations.
In lifted system form (where for reasons of clarity only the x-direction is considered), the ILC with nonlinear Q-filter is represented by the block diagram of Figure 5. Here
- sx(k) Lx 6 z −1I - i− + - -−Sc,x - S x -i - Qx(·) 6 dx(k) + ex(k) −
Fig. 5. Long-stroke wafer stage iterative learning control scheme in x-direction.
Sx ∈ Rn×n and Sc,x∈ Rn×n represent the sensitivity and complementary sensitivity matrix, respectively, in Toeplitz form, dx(k) ∈ Rn represents a column with output distur-bances corresponding to the k-th trial and ex(k) ∈ Rn the corresponding servo error signals in x-direction. Without any further detail, the nonlinear Q-filter matrix is represented by
Qx(ex(k)) ∈ Rn×n. The learning gain matrix Lx∈ Rn×nis the pseudo-inverse of the complementary sensitivity matrix, or Lx = (ST
c,xSc,x+ λxI)−1STc,x (see [8]) with λx = 1 a tuning parameter used to dominate stability threatening (and often badly scaled) singular values in ST
c,xSc,x. The set-point correction sx(k) ∈ Rn which results from the learning scheme is used to shape the input r(t) in Figure 2 as
to reduce (and ultimately eliminate) any recurring (settling) behavior from the error responses.
From Figure 5, it follows that
ex(k) = Sc,xsx(k) + Sxdx(k), (5) and sx(k + 1) = sx(k) − LxQx(ex(k))ex(k) = sx(k) − LxQx(ex(k))(Sc,xsx(k) + Sxdx(k)) = (I − LxQx(ex(k))Sc,x)sx(k) − LxQx(ex(k))Sdx(k), (6) hence the first update law reflecting the learned set-point corrections sx. The second update law (reflecting the learned servo errors ex) is given by
ex(k + 1) = Sc,xsx(k + 1) + Sxdx(k + 1) = Sc,x(I − LxQx(ex(k))Sc,x)sx(k) − Sc,xLxQx(ex(k))Sxdx(k) + Sxdx(k + 1) = (I − Sc,xLxQx(ex(k)))ex(k) + Sx(dx(k + 1) − dx(k)). (7) Apart from the x-direction, for the long-stoke wafer stage such as considered in the previous section, the learning scheme in Figure 5 and the update laws in Eqs. (6) and (7) are simultaneously applied to the y− and rz-directions, respectively. The validity of such an approach intuitively requires the coupling between the subsequent directions to be sufficiently small. A practical way of dealing with such
coupling is to reduce the learning gains but with the effect that more learning trials are required, each trial potentially inducing noise in the updated learning signals.
To avoid the amplification of noise through learning, a nonlinear Q-filter structure, see Figure 6, is adopted. Given
- 2 - i - Q x - i - φ(ex) ? 6 + -+ -ex e˜x
Fig. 6. Nonlinear Q-filter control scheme.
the low-pass filter characteristics of the linear Q-filter part:
Qx(s) =
ωlp,x
s+ ωlp,x
, (8)
with cut-off frequency ωlp,x = 3000π rads−1, a nonlinear Q-filter operation is obtained by feeding back a nonlinear gain φ(·). This nonlinear gain is defined as
φ(ex(t)) = α, if|ex(t)| ≤ δ, αδ |ex(t)| , if |ex(t)| > δ, (9)
with δ = 5µm a saturation band and α = 20 a switching
gain. The effect of (9) is shown in the left part of Figure 7. For large signal levels ex, hence the kind of levels
1 2500 −50 0 0 0 ex in µm φ (ex ) -δ δ α frequency in Hz m ag n it u d e in d B
Fig. 7. Nonlinear Q-filter characteristics.
corresponding to a large (and preferable) signal-to-noise ratio, the switching gain φ(ex) is small. Contrarily, for small signal levels having a small (and non-preferable) signal-to-noise ratio, the switching gain φ(ex) is large. The effect in terms of the nonlinear Q-filter operation is shown in the right part of Figure 7. Namely, from Figure 6 it follows that
L{˜ex(t)} = Qx(s)L{2ex(t) − ˜ex(t)}
− L{φ(ex(t))˜ex(t)}.
(10) For the special case that φ(ex(t)) in (9) equals a constant:
φ(ex(t)) = 0 (upper limit) and φ(ex(t)) = α (lower limit),
the following Q-filter transfer limits are obtained:
˜ex(s) ex(s) upper limit = 2Qx(s) 1 + Qx(s) , and ˜ex(s) ex(s) lower limit = 2Qx(s) 1 + α + Qx(s) . (11)
These limits form the boundary characteristics in the right part of Figure 7. The band of low-pass filter characteristics in the nonlinear Q-filter follows from the fact that through (9) each value0 ≤ φ(ex(t)) ≤ α can be assessed. Note that the choice forQx being a first-order causal filter is exemplary and not intended to be restrictive.
IV. LYAPONOVSTABILITYANALYSIS
In the more general context of an error vector e ∈ Rn, a sensitivity matrix S ∈ Rn×n and a vector d ∈ Rn containing output disturbances, stability of the nonlinear learning control such as derived in Section III can be studied using the following Lyapunov function candidate
V(k) = (e(k) − Sd(k))T
(e(k) − Sd(k)), (12) whose variation is given by
V(k + 1) − V (k) = −eT(k)(AT(e(k)) + A(e(k)))e(k)
+ eT(k)AT(e(k))A(e(k))e(k)
+ 2dT(k)STA(e(k))e(k),
(13) with A(e(k)) = ScLQ(e(k)) and Sc, L, Q∈ Rn×n. For the given choice of Q(e(k)) it can be deduced that there exist
a positive definite matrix P satisfying
xTPx≤ xTAT(e(k))A(e(k))x ≤12xT(AT
+ A)x ≤ xTS
cLx.
(14)
It therefore follows that for all d(k) = d(k + 1) = 0, (13)
can be upper bounded by
V(k + 1) − V (k) ≤ −eT(k)Pe(k). (15) Hence a negative definite function for any e(k) 6= 0. This combined with (12) being a positive definite function renders global asymptotic stability of the origin, or k→∞lim e(k) = 0.
For the case with disturbances, i.e., d(k) = d(k + 1) 6= 0, it follows that
V(k + 1) − V (k) ≤ −keT
(k)kP(keT(k)kP− ∆η), (16) with keT(k)k
P = peT(k)Pe(k) a P-norm on e(k), ∆ =
δ√n with δ a uniform bound on each element in d, and η
the smallest singular value of STA+ ATS. As a result, (16)
renders
ǫ= {e(k) ∈ Rn
,ke(k)kP≥ ∆η} , (17) i.e., a positively invariant set to which all solutions converge. Since e(k) is related to s(k) via (5), it follows that if e(k)
converges also s(k) converges given the properties of S and d(k). In fact, both converge to a ball about 0 in finite time.
V. LONG-STROKEWAFERSTAGEPERFORMANCE
The effectiveness of the nonlinear learning control in achieving improved performances on a long-stroke wafer stage is assessed in three parts. First, the translation from an updated (and learned) error signal to an updated set-point representation is discussed. This translation is necessary in view of the limited long-stroke wafer stage application pos-sibilities of injecting a signal update. Second, the nonlinear learning is evaluated in the presence of noise and uncertainty. Third, the ability of the learning control to obtain improved long-stroke wafer stage settling behavior is demonstrated. A. From updated error to updated set-point
In applying the updated error signals from learning, the long-stroke wafer stage supports the means to inject extra polynomial-based set-point signals. Prior to this approxima-tion, however, first the updated error signals s are translated to set-point signal updates ∆r. This translation (see also
Figure 2) is given in Laplace domain by
∆r(s) = (Cff(s) + Cfb(s))−1Cfb(s)s(s). (18) For the x-direction, the effect of the polynomial approxi-mation of ∆rx by ∆ ˜rx is shown in Figure 8. The upper
0 0.5 0 1 0 0.5 −0.5 0 0.5 0 0.5 −40 0 40 time in s ∆ rx an d ∆ ˜rx in µ m ∆ rx − ∆ ˜rx in n m
non-equidistant 64-point grid
Fig. 8. Time-series comparison between updated set-point signal ∆rx
and polynomial-based approximation ∆ ˜rx (middle part) along with the
approximation error∆rx− ∆ ˜rx(lower part) and the non-equidistant
64-point grid (upper part).
part shows the grid used in the approximation of the learned signals. In the middle part, both the learned set-point signal
∆rx (in gray) and its polynomial-based approximation∆ ˜rx (in black) are depicted. In each of the 64 intervals the set-point is approximated by a fourth-order polynomial satisfy-ing continuity and differentiability at the boundaries of the interval, see also [1]. In the lower part, the error between the set-point update and the polynomial approximation is shown. Note that the choice for the grid (though rather arbitrary) relates to the indicated area where performance in terms of reduced settling times should be obtained.
B. Learning in the presence of noise and uncertainty The aim of the nonlinear Q-filter is to reduce the effect of noise in the learning update. The latter being the result of a generally large number of learning trials as to compensate for the small-valued accessible learning gains. This is best shown in simulation by considering 100 trials where the learned updates s(k) after each trial k are corrupted with
a random noise vector d(k), the result of which is depicted
in Figure 9. For four values of the saturation length δ ∈
0 100 100 number of trials k L2 -n o rm o n erz (k ) in m il li ra d δ= 0µm δ= 1.67µm δ= 5µm δ= 15µm
Fig. 9. Time-series comparison for different nonlinear Q-filter settings
δ∈ {0, 1.67, 5, 15} µm and ωlp,x=3000π[rads−1].
{0, 1.67, 5, 15} µm, see also (9), and through the L 2-norm on the error erz, this figure shows that larger values for δ which (on average) correspond to stronger low-pass filter properties induce less noise in the learning updates and therefore correspond to smaller error levels. Naturally the advantage comes particularly to the fore after a sufficient number of trials needed for the error levels to become suffi-ciently small and hence the nonlinear Q-filtering to become sufficiently active. Furthermore, the significant reduction in noise sensitivity seems to come at the limited cost of convergence speed. This is because large (and incidentally occurring) signal levels still apply for limited low-pass filtering. The choice for δ though practically determined by
0 0.5 −400 0 400 0 0.5 −5 0 5 time in seconds erz in µ ra d erz (i )-1 5 P 5 j= 1 erz (j ) in µ ra d
Fig. 10. Time-series measurement of the erz errors of five consecutive
motions along with the difference between each of these signals erz(i) with i∈ {1, 2, . . . , 5} and the mean of these signals: 1
5 P5
j=1erz(j).
noises. To access these noises, Figure 10 shows the result of five consecutive error signals resulting from a similar set-point motion profile which is sequentially applied to the wafer stage system. In the lower part of the figure, the noise signal in each of these trials is determined by the difference between the error signal at one particular trial and the mean error signal over all five trials. It can be seen that the so-obtained noise signals validate a saturation band of 5 µrad. C. Improved settling behavior through learning
Improved settling behavior through learning is demon-strated in Figure 11. For the errors ex, ey, and erz it can
0 0.5 −400 0 400 0 0.5 −400 0 400 0 0.5 −400 0 400 time in seconds ex in µ m ey in µ m erz in µ ra d ← controller window → ← controller window → ← controller window →
Fig. 11. Time-series measurement of the errors ex, ey, and erzusing a
learning controller window as to avoid actuator saturation.
be seen that beyond the dictated set-point profiles a smaller settling phenomenon (thick curves) results from learning than prior to learning (thin curves). During the set-point profile, no learning is applied. This is because of the implementation through polynomial approximations which causes significant acceleration contributions. Hence the kind of contributions that atop of the set-point accelerations renders severe actuator saturation. To avoid such saturation, a controller window is defined (the indicated interval) in which the learning process is allowed to generate updated control signals [7]. Outside the controller window the updated signals are being forced to zero. Though the settling phenomenon is still partly present because of the controller window limitations, the settling times needed for the errors to become sufficiently small is reduced significantly which, in view of the noises present, validates the method.
VI. CONCLUSIONS
A nonlinear Q-filter is proposed as a means to reduce the transmission of noise through learning in schemes requiring a sufficiently large number of learning trials. Given the am-plitude characteristics of the input signals to the learning, the Q-filter acts as a low-pass filter with a large cut-off frequency (hence fast learning but limited robustness properties) for
large amplitude signals. For small amplitude signals the cut-off frequency is significantly reduced (through reduction of the static filter gain) thereby avoiding the transmission of noise through learning. Stability of the nonlinear learning is guaranteed using Lyapunov analysis whereas performance is demonstrated on a long-stroke wafer stage of a wafer scanner. Here a significant improvement in settling behavior is obtained despite the constrained conditions under which the learning algorithm has been implemented.
REFERENCES
[1] Atsumi T. (2009) Feedforward control using sampled-data polynomial for track seeking in hard disk drives. IEEE Transactions on Industrial Electronics, 56(5):1338-1346.
[2] Baggen M, Heertjes MF, and Kamidi R. (2008) Data-based feed-forward control in MIMO motion systems. In Proceedings of the American Control Conference, Seattle, WA:3011-3016.
[3] Bode CA, Ko BS, and Edgar TF. (2004) Run-to-run control and performance monitoring of overlay in semiconductor manufacturing. Control Engineering Practice, 12: 893-900.
[4] Bristow DA, Alleyne AG, and Tharayil M. (2008) Optimizing learn-ing convergence speed and converged error for precision motion control. Journal of Dynamic Systems, Measurement, and Control, 130(5):054501.
[5] Cai Z, Freeman CT, Lewin PL, and Rogers E. (2008) Iterative learning control for a non-minimum phase plant based on a reference shift algorithm. Control Engineering Practice, 16:633-643.
[6] Chen YQ, Moore KL, Yu J, and Zhang T. (2008) Iterative learning control and repetitive control in hard disk drive industry - a tutorial. International Journal of Adaptive Control and Signal Processing, 22(4):325-343.
[7] Dijkstra BG, and Bosgra OH. (2002) Extrapolation of optimal lifted system ILC solution, with application to a waferstage. In Proceedings of the American Control Conference, Anchorage, AK:2595-2600. [8] Ghosh J, and Paden B. (2002) A pseudoinverse-based iterative learning
control. IEEE Transactions on Automatic Control, 47(5):831-837. [9] Hätönen J, Owens DH, and Feng K. (2006) Basis functions and
param-eter optimisation in high-order iterative learning control. Automatica, 42(2):287-294.
[10] Heertjes MF, and Tso T. (2007) Nonlinear iterative learning control with applications to lithographic machinery. Control Engineering Prac-tice, 15:1545-1555.
[11] Heertjes MF, and Van de Molengraft MJG. (2009) Set-point variation in learning schemes with applications to wafer scanners. Control Engineering Practice, 17:345-356.
[12] Longman RW. (2000) Iterative learning control and repetitive control for engineering practice. International Journal of Control, 73(10):930-954.
[13] Mishra S, Coaplen J, and Tomizuka M. (2007) Precision positioning of wafer scanners; segmented iterative learning control for nonrepetitive disturbances. IEEE Control Systems Magazine, 8:20-25.
[14] Van der Meulen, SH, Tousain R, and Bosgra OH. (2008) Fixed structure feedforward controller design exploiting iterative trials: ap-plications to a wafer stage and a desktop printer. Journal of Dynamic Systems, Measurements, and Control, 130(5):051006.
[15] Phan M, and Longman RW. (1988) A mathematical theory of learning control for linear discrete multivariable systems. In Proceedings of the AIAA/AAS Astrodynamics Specialist Conference, Minneapolis, MN: 740-746.
[16] Steinbuch M, Weiland S, and Singh T. (2007) Design of noise and period-time robust high-order repetitive control, with application to optical storage. Automatica, 43:2086-2095.
[17] Wijdeven JJM van de. (2008) Iterative learning control design for un-certain and time-windowed systems. PhD-thesis, Eindhoven University of Technology, The Netherlands.
[18] Xu J, Sun M, and Yu L. (2005) LMI-based robust iterative learning controller design for discrete linear uncertain systems. Journal of Control Theory and Applications, 3:259-265.
