102 ieee transactions on control systems … · 102 ieee transactions on control systems...

102 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 22, NO. 1, JANUARY 2014

Connecting System Identification andRobust Control for Next-Generation Motion

Control of a Wafer StageTom Oomen, Robbert van Herpen, Sander Quist, Marc van de Wal, Okko Bosgra, and Maarten Steinbuch

Abstract— Next-generation precision motion systems are light-weight to meet stringent requirements regarding throughputand accuracy. Such lightweight systems typically exhibit lightlydamped flexible dynamics in the controller cross-over region.State-of-the-art modeling and motion control design proceduresdo not deliver the required model complexity and fidelity tocontrol the flexible dynamical behavior. The aim of this paper isto develop a combined system identification and robust controldesign procedure for high performance motion control and applyit to a wafer stage. Hereto, new connections between systemidentification and robust control are employed. The experimentalresults confirm that the proposed procedure significantly extendsexisting results and enables next-generation motion controldesign.

Index Terms— Control applications, model uncertainty, modelvalidation, motion control, motion systems, multivariable control,robust control, system identification for control.

I. INTRODUCTION

A. Developments in Lithography

The mass production of integrated circuits (ICs) has enabledthe development of a wide variety of technologies that havea key role in today’s society, including transportation sys-tems, manufacturing systems, personal computers, and mobilephones. Wafer scanners (Fig. 1) are the state-of-the-art equip-ment for the automated production of ICs. During the produc-tion process, a photoresist is exposed on a silicon disc, calleda wafer. During exposure, the image of the desired IC patterns,which is contained on the reticle, is projected through a lenson the photoresist. The exposed photoresist is then removedby means of a solvent. Subsequent chemical reactions enableetching of these patterns, which is repeated for successivelayers. Typically, more than 20 layers are required for eachwafer. Each wafer contains more than 200 ICs that are sequen-tially exposed. During this entire process, the wafer must

Manuscript received June 10, 2011; revised October 31, 2012; acceptedJanuary 31, 2013. Manuscript received in final form February 4, 2013.Date of publication March 6, 2013; date of current version December 17,2013. This work was supported by Philips Applied Technologies, Eindhoven,The Netherlands. Recommended by Associate Editor G. Cherubini.

T. Oomen, R. van Herpen, S. Quist, O. Bosgra, and M. Steinbuchare with the Eindhoven University of Technology, Eindhoven 5600MB, The Netherlands (e-mail: [email protected]; [email protected];[email protected]; [email protected]; [email protected]).

M. van de Wal was with the Mechatronics Department, PhilipsApplied Technologies, Eindhoven 5600 MB, The Netherlands (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCST.2013.2245668

Fig. 1. Schematic illustration of a wafer scanner system, where ➀ lightsource, ➁ reticle, ➂ reticle stage, ➃ lens, ➄ wafer, and ➅ wafer stage.

extremely accurately track a predefined reference trajectory insix motion degrees-of-freedom (DOFs). This precision motiontask is performed by the wafer stage, which is investigated indetail in this paper.

In the last decades, increasing demands with respect tocomputing power and memory storage have led to an ongoingdimension reduction of transistors. The minimum feature sizeassociated with these transistors is called the critical dimension(CD) and is determined by the wavelength of light, see [1], [2].In [2], CDs of 50 nm have been achieved using deep ultra-violet (DUV) light with a wavelength of 193 nm throughmany enhancements of the production process. However, atechnology breakthrough is required to reduce the wavelengthof light and consequently improve the achievable CD.

Extreme ultraviolet (EUV) is a key technology fornext-generation lithography [3], [4]. At present, experimen-tal prototypes with a 13.5 nm wavelength are reported in[2] and [5] and the first production systems are presently beinginstalled [6].

1063-6536 © 2013 IEEE

OOMEN et al.: SYSTEM IDENTIFICATION AND ROBUST CONTROL FOR MOTION CONTROL 103

The introduction of EUV light sources in lithography hasfar-reaching consequences for all subsystems of the waferscanner, including the wafer stage. EUV does not transmitthrough any known material, including air. Hence, lenses usedin DUV have to be replaced by mirrors. Moreover, the entireexposure has to be performed in vacuum.

B. Developments in Precision Motion Systems

Due to the developments in lithographic productionprocesses, next-generation precision motion systems areexpected to be lightweight for several reasons. First, vacuumoperation requires these systems to operate contactless toavoid pollution due to mechanical wear or lubricants. Inaddition, contactless operation reduces parasitic nonlinearities,such as friction and thus potentially increases reproducibility.Since contactless operation requires a compensation of gravityforces, a lightweight system is essential. Second, marketviability requires a high throughput of the wafer scanner.This requires high accelerations in all six motion DOFs. Theaccelerations a of the wafer stage are determined by Newton’slaw F = ma. Here, the forces F that the electromagneticactuators can deliver are bounded, e.g., due to size require-ments and thermal reasons, and are proportional to a2. Hence,a high acceleration a requires a reduction of the mass m,again motivating a lightweight system design. Third, the waferdiameter is expected to increase from 300 to 450 mm toincrease productivity. This requires increased dimensions ofthe wafer stage, which again underlines the importance of alightweight system design.

As a result of a lightweight system design, next-generationmotion systems predominantly exhibit flexible dynamicalbehavior at lower frequencies. This has important conse-quences for control design, as is investigated next.

C. Toward Next-Generation Motion Control: The Necessity ofa Model-Based Approach

On the one hand, the increasing accuracy and performancedemands lead to the manifestation of flexible dynamicalbehavior at lower frequencies. On the other hand, due tothese increasing demands, the controller has to be effectiveat higher frequencies. Combining these developments leads toa situation where flexible dynamical behavior is present withinthe control bandwidth. This is in sharp contrast to traditionalpositioning systems where the flexible dynamical behavior canbe considered as high-frequency parasitic dynamics, as is, e.g.,the case in [7, Sec.. 2.1, Assumptions 1–3].

The presence of flexible dynamical behavior within thecontrol bandwidth has significant implications for motioncontrol design in comparison to the traditional situation asfollows.

1) Next-generation motion systems are inherently multi-variable, since the flexible dynamical behavior is gener-ally not aligned with the motion DOFs.

2) Next-generation motion systems are envisaged tobe designed with many actuators and sensors toactively control flexible dynamical behavior, whereas

traditionally the number of inputs and outputs equalsthe number of motion DOFs.

3) A dynamical relation exists between measured and per-formance variables, since the sensors generally measureat the edge of the wafer stage system, while the perfor-mance is required on the spot of exposure on the waferitself. In contrast, the flexible dynamical behavior isoften neglected in traditional motion systems, leading toan assumed static geometric relation between measuredand performance variables.

These implications of lightweight motion systems on thecontrol design motivate a model-based control design, since:

1) a model-based design provides a systematic controldesign procedure for multivariable systems;

2) a model is essential to investigate and achieve the limitsof performance. Specifically, fundamental performancelimitations are well-established for nominal models,see [8], and robust control provides a transparent tradeoffbetween performance and robustness, see [9];

3) a model-based design procedure enables the estimationof unmeasured performance variables from the measuredvariables through the use of a model.

As pointed out in [7], a model-based control design is farfrom standard in state-of-the-art industrial motion control,since the majority of such systems is controlled by manually-tuned single-input single-output (SISO) proportional-integral-derivative (PID) controllers in conjunction with rigid-bodydecoupling based on static input–output transformations.

D. Modeling for Precision Motion Control

The success of a model-based control design hinges on themodel quality. For the considered class of motion systems,system identification provides an inexpensive, fast, and accu-rate methodology to obtain a model. This is especially dueto the fact that these motion systems are designed such thatthe system dynamics are essentially linear, enabling the useof well-developed system identification techniques for linearsystems. The resulting linear model is an approximation of thetrue system, since:

1) motion systems generally contain many resonancemodes [10] of which a limited number is included inthe model;

2) parasitic nonlinearities are present, e.g., nonlinear damp-ing [11];

3) identification experiments are based on finite time dis-turbed observations of the true system.

Robust control design [9], [12], explicitly addresses thesemodel errors by considering a model set that encompassesthe true system behavior. This model set has to be chosenjudiciously, since the resulting robust controller has to achieveperformance for all candidate models in this set in the sense ofan appropriately defined control objective. This leads to tworequirements on the model set.

R1) The model set should enable a high performance robustcontrol design, i.e., all candidate models should achievehigh performance with a single controller.


R2) The model set should enable a nonconservative syn-thesis of a robust controller that has a low-order forreal-time implementation.

In Requirement 2, low-order refers to the order of magnitudeof the number of states, where 101 and 102 states are typicallyconsidered low-order for multivariable motion applications.

In [13], a combined system identification and robustcontrol design approach for SISO systems are presented. Thisapproach is further extended toward multi-input multioutput(MIMO) motion systems in [7], whereas related approachesfor lightly damped mechanical structures are presented in[14] and [15]. However, as is also argued in [7], the achievableperformance for MIMO systems is hindered by inadequaciesin the system identification procedure.

In [16], a first important step is taken to connect the systemidentification criterion and the control criterion. However,similar to the approach in [7], the actual procedure involvesthe use of highly structured model uncertainty descriptions.As a result, the robust controller synthesis step is generallyconservative since it is based on upper bounds that are knownto be conservative for highly structured model uncertaintydescriptions [17, Sec. 9]. Hence, such an approach violatesRequirement 2, above. The aim of this paper is to develop anapproach that employs unstructured uncertainty during systemidentification in a nonconservative manner. As a result, robustcontrol can also directly be performed in a non-conservativemanner. Thus, such an approach extrapolates well tonext-generation motion control applications where a highnumber of inputs and outputs are expected, which is in sharpcontrast to existing approaches, including [7], [16].

E. Contribution and Outline of This Paper

The main contribution of this paper is the development andimplementation of a joint system identification and robust con-trol design framework for high performance next-generationmotion control that satisfies requirements 1) and 2) inSection I-D. Hereto, a design framework is proposed thatexploits a new connection between system identification androbust control. The consequence of the resulting model set isthat it allows both nonconservative model set identification andnonconservative robust controller synthesis. In this respect, theapproach significantly extends earlier contributions in systemidentification and robust control for motion systems, including[7], [13], [14], [16].

In this paper, theoretical, design, and algorithmic aspects areappropriately addressed to obtain a practically implementablecontrol design procedure for next-generation motion systems.The resulting framework is experimentally demonstrated onan industrial wafer stage system. The outline of this paperis as follows. In Section II, the considered wafer stage isintroduced and the wafer stage control problem is stated.Then, in Section III, the general framework that connectssystem identification and robust control for wafer stage motionis presented. In Section IV, a procedure for identificationof a wafer stage model for robust control is presented andapplied to the experimental wafer stage system. This model issubsequently used in Section V to design a robust controller

Fig. 2. Experimental wafer stage system, where ➀: metrology frame, ➁mover, and ➂ airmount. Only the wafer stage is shown, i.e., without theimaging optics in Fig. 1.

Fig. 3. Close-up of experimental wafer stage system in Fig. 2, where➀ mirror block, ➁ mover, ➂ guardrail, ➃ current coil, and ➄ magnet stator.

and to implement it on the wafer stage system. Conclusionsand a discussion are provided in Section VI.

II. WAFER STAGE CONTROL PROBLEM

A. Experimental Setup

The considered industrial wafer stage system is especiallydesigned for vacuum operation and is depicted in Figs. 2 and 3.The system is equipped with moving-coil permanent magnetplanar motors that enable contactless operation, see [18] forthe underlying principle. The motion system consists of twoparts: a stator, which is a plate consisting of an ordered arrayof permanent magnets, and a mover, which constitutes themoving part of the wafer stage.

Four actuators are connected to the mover to provide thenecessary force, each consisting of three coils, which arepowered by a three-phase power source. By means of anappropriate position-dependent commutation of the coils, eachactuator delivers a well-defined and reproducible force in twodirections. As a result, eight independent forces are available.


Fig. 4. Identified frequency response function ˜Po(ωi ), ωi ∈ �id of waferscanner.

Laser interferometers in conjunction with a mirror block,which are connected to the metrology frame and the waferstage, respectively, enable a high accuracy position measure-ment in all six motion DOFs, i.e., three translations and threerotations. Specifically, subnanometer measurement accuracy isavailable for the translational DOFs. Throughout, all signalsand systems operate in discrete time with a sampling frequencyof 2.5 kHz. Consequently, the performance criteria are definedin discrete time. In addition, the entire system identificationand controller synthesis are performed in discrete time. Forextensions of the presented framework toward sampled-dataaspects, including intersample behavior, see [19].

It is emphasized that the presented approach in this paperis aimed to deal with a large number of actuators and sensors,possibly more than the number of motion DOFs. However, tofacilitate a clear exposition, the controller design in this paperis performed for a two-input two-output subsystem. The otherDOFs are controlled by low performance PID controllers.The translational x and y DOFs1 in the horizontal plane areconsidered in this paper. Hence, the input u and output y tothe system are given by

u =[

ux

uy

]

, y =[

yx

y y

]

1The variable y is used to denote both a translational direction and measuredvariable, it should be clear from the context which one is referred to.

Fig. 5. Generic wafer stage feedforward and feedback control problem.

whereas the system is partitioned as

P =[

Px x Pxy

P yx P yy

]

.

An identified frequency response function ˜Po(ωi ), ωi ∈ �id

is depicted in Fig. 4. This frequency response function isobtained using the approach in Appendix A, and �id is asuitably chosen discrete frequency grid. From Fig. 4, thefollowing observations are made. Below 200 Hz, the systemis decoupled, revealing a rigid-body behavior in the diagonalelements, corresponding to translations in the x-directionand y-direction, respectively. The first resonance phenomenaappear at 208 and 214 Hz in all elements of ˜Po(ωi ). Sincethese flexible dynamics are not aligned with the motion DOFs,the interaction between the x-direction and y-direction is high,i.e., the four elements of ˜Po(ωi ) have an approximately equalgain beyond 200 Hz.

B. Wafer Stage Control Goal

1) Bandwidth Definition: The key performance indicatorfor wafer stage systems is the closed-loop bandwidth. Theconsidered wafer stage system in this paper is decoupled up toapproximately 200 Hz, see Section II-A. In case, the controller

C =[

Cx x Cxy

C yx C yy

]

is (approximately) diagonal in the low frequency range,then the bandwidth can be defined for the x-direction andy-direction separately. In this paper, the bandwidth for eachseparate direction is defined as the gain crossover frequencyfBW [9, Section 2.4.3], which is defined as the smallestfrequency for which

|Px x Cx x | = 1 and |P yyC yy| = 1

for the x-direction and y-direction, respectively. By diagonal-ity of P and C , this in turn corresponds to the frequencieswhere the singular values σ(PC) equal one and hence σ(C P)equal one.

2) Motivation for Bandwidth Specifications: The concept ofbandwidth as defined in Section II-B1 is particularly useful forwafer stages that have to perform a servo task, i.e., to tracka predefined trajectory with high accuracy. Assuming that theperformance variables are measurable for feedback, the gen-eral control problem can be represented as in Fig. 5. Herein,r2 is the reference signal that should be accurately tracked,i.e., e = r2 − y should be kept small, where y is the systemoutput. In addition, r1 has the role of feedforward commandsignal, whereas d1 and d2 represent unknown disturbances atthe system input and output, respectively. Finally, v representsmeasurement noise.


To illustrate the notion of bandwidth, notice that directmanipulations reveal that the servo error e equals

e = So(r2 − Por1) − Sod − Sov (1)

with So = (I + PoC)−1 and d = d2 + Pod1. For the consideredwafer stage system, analysis of the three terms in (1) leads tothe following observations.

1) The term So(r2 − Por1) can be made small by anappropriate feedforward control design, i.e., setting r1 =Fr2 with F ≈ P−1

o . Suitable techniques to design Ffor motion systems are available in [20] and referencestherein. For the considered class of motion systems, theresulting setpoint-induced error (I − Po F)r2 generallyhas dominant frequency content in the low-frequencyrange.

2) The disturbance term d generally has dominant low-frequency content.

3) Due to sub-nanometer measurement accuracy, the mea-surement noise v is orders of magnitude smaller than theeffect of the disturbance d and setpoint-induced errorand is typically negligible.

After appropriate feedforward design F and neglectingmeasurement noise v, the resulting error signal is then

e = So(I − Po F)r2 − Sod.

The aim of this paper is to design a feedback controller Cthat leads to a small error signal. Hereto, observe that by theresults of [21]

σ (So) ≈ 1

σ (PoC)(2)

for frequencies where σ (PoC) � 1. In (2), σ (.) and σ(.) referto the largest and smallest singular value, respectively.

By noting that increasing the bandwidth increases the fre-quency range for which σ(PoC) � 1 and increases σ(PoC)at low frequencies, it is observed from (2) that σ (So) isreduced in that case. Hence, additional disturbance attenuationregarding the setpoint-induced error (I − Po F)r2 and thedisturbance d is achieved at low frequencies.

3) Bandwidth Specification: High bandwidth is desirable forhigh performance motion control. However, flexible dynamicsand their uncertainty generally impose an upper bound on theachievable bandwidth. Hence, robustness should be appropri-ately taken into account. In this paper, robustness is addressedby means of a formal robust controller design in the nextsection.

When designing a manually tuned PID controller withoutusing any notch filters, the bandwidth is limited to approx-imately 40 Hz. The PID controller that achieves the 40 Hzbandwidth while maintaining good robustness margins isdenoted Cexp (Fig. 15).

The aim of this paper is to achieve a bandwidth of 90 Hz.To anticipate on the next section, the presented robust con-trol design framework requires the specification of a targetbandwidth. The controller synthesis procedure then aims toachieve this target bandwidth. The motivation for selecting abandwidth equal to 90 Hz is twofold. On the one hand, theexogenous disturbances have a dominant frequency content

Fig. 6. Feedback configuration for wafer stage application.

below approximately 100 Hz (Fig. 19). This motivates abandwidth of at least 100 Hz. On the other hand, it is knownfrom practical experience that such a bandwidth is difficult toobtain due to the presence of resonance phenomena around200 Hz. Hence, a bandwidth of 90 Hz is challenging.

In the next section, a model-based optimal controller designis pursued for achieving the bandwidth requirements thatenable the direct design of a multivariable controller that takesinto account the inherent coupling in the system. In addition,by explicitly taking model uncertainty into account, a robustcontroller can be designed that provides certain performanceguarantees when implemented on the true system. Noticethat an alternative design approach could be to extend amanually tuned PID controller with notch filters to suppressthe resonance phenomena around 200 Hz. However, this is notstraightforward due to the inherent coupling in the system.

III. GENERIC IDENTIFICATION AND ROBUST

CONTROL FRAMEWORK

A. Optimal Controller Design

1) Control Criterion: In Section II-B, it is argued that thewafer stage control goal imposes requirements on the loop-gain in terms of the bandwidth. A systematic manner to attainthese requirements is by formulating a certain criterion usingthe H∞-norm.

In this paper, the criterion

J (P, C) = ‖W T (P, C)V ‖∞ (3)

is considered, where the goal is to compute the optimal controldesign given by

Copt = arg minC

J (Po, C). (4)

Besides the fact that the H∞-norm enables the specificationof requirements on the loop-gain, including bandwidth, theH∞-norm in (3) has important advantages compared to alter-native system norms. First, the H∞-norm is an induced normand enables a suitable representation of model uncertainty asH∞-norm bounded perturbations. This property will be usedto formulate robustness objectives in Section III-B. Second,the required synthesis algorithms are commercially available.

To specify the criterion (3), the feedback interconnection inFig. 6 is considered. As a result, T (P, C) is defined as

T (P, C):

[

r2r1

]

�→[

yu

]

=[

PI

]

(I + C P)−1 [

C I]

(5)

and the weighting functions are partitioned accordingly as

W =[

Wy 00 Wu

]

, V =[

V2 00 V1

]


(a) (b)

Fig. 7. (a) Singular values of weighting filters: W1 (solid blue), W2 (dashedred). (b) Singular values of the nonparametric frequency response functionestimate ˜Po(ωi ) for ωi ∈ �id (solid blue), shaped system W2˜Po(ωi )W1 forωi ∈ �id (dashed red).

The four-block problem (5) guarantees internal stability ofthe resulting feedback loop and is sufficiently general toencompass many common robust control design approaches,including the approach in Section III-A2 that specifies theloop-gain in terms of bandwidth requirements.

2) Loop-Shaping Weighting Functions W and V : Inthis paper, a loop-shaping-based weighting function designapproach is adopted that resembles the approach in[21] and [22]. Such a loop-shaping approach can be effec-tively used to specify the bandwidth requirements for motionsystems, as is also evidenced by the related successful motioncontrol design applications in [7], [13], [16], and [23].

In the design approach of [21] and [22], weighting filters W2and W1 are specified such that W2 PW1 has a certain desiredopen-loop shape PC . Note that prior applications of the loop-shaping procedure in [21] and [22] require knowledge ofthe nominal parametric model P for designing the weightingfilters. Since such a model is not yet available, it is proposedto employ the identified frequency response function estimate˜Po(ωi ), see Fig. 4.

To specify the open-loop shape, note that σ (PoC) has to bemade large at low frequencies in view of (2). Besides settingthe bandwidth fBW, as defined in Section II-B1, equal to 90 Hzfor both the x-direction and the y-direction, additional low-frequency disturbance attenuation can be achieved. Hereto,W1 is designed to enforce integral action to attenuate low-frequency disturbances. The cut-off frequency for the integralaction is set to fBW/5, see Fig. 7.

A bound similar to (2) can be derived to make σ (PC)small at high frequencies, which effectively attenuates the(small) contribution in (1) due to v at high frequencies. Hereto,W2 is designed to enforce controller roll-off to attenuate high-frequency measurement noise. In addition, W2 is designed tohave a slope of approximately +1 around the target such thatthe loop-gain W2 PW1 has a gain of approximately −1. Thisfacilitates achieving good gain and phase margins. Finally,the multivariable gain of the shaped system is adjusted usinga modified align algorithm, see [24], such that the singularvalues equal unity at the target bandwidth fBW = 90 Hz. Asa result, the weighting filters W1 and W2 are non-diagonaltransfer function matrices. The designed weighting filters andthe resulting desired loop-shape W2 ˜PoW1 are depicted inFig. 7.

Finally, direct algebraic computations reveal that thedesigned weighting filters W1 and W2 correspond to W andV in (3) as

W =[

W2 00 W−1

1

]

, V =[

W−12 00 W1

]

.

Note that for the controller synthesis, the weighting filters canequally be absorbed into the feedback loop as in [21] and [22]to enable the use of standard H∞-optimization algorithms,e.g., [12], that require bistable weighting functions.

The criterion J (P, C) is an indicator showing whetherthe designed loop-gain PC mimics the specified loop-gainW2 PW1, as is proved by the bounds in [21]. For instance,a common guideline in such loop-shaping techniques is thatthe desired loop-shape is accurately matched if J (P, C) < 4.In this paper, the criterion J (P, C) is considered an auxiliaryvalue in the sense that a smaller value implies that the targetbandwidth is achieved closer.

B. Robustness Specification

The optimal controller in (4) cannot be computed directlysince Po is unknown. To perform the actual controller syn-thesis, a parametric model of the system is required. As isargued in Section I-D, enforcing robustness is essential forsuch model-based control design approaches. However, thecriterion J (P, C) in (6) only reflects performance objectivesand does not necessarily guarantee robustness with respect tomodel errors.

In view of robustness requirements, a model set P is con-sidered. This model set is constructed such that it encompassesthe true system behavior, i.e., it satisfies

Po ∈ P . (6)

Throughout, the model set P is constructed by considering aperturbation �u around a nominal model P

P ={

P∣

∣P = Fu(H ,�u),�u ∈ �u

}

(7)

where the upper linear fractional transformation (LFT) is givenby

Fu(H ,�u) = H22 + H21�u(I − H11�u)−1 H12.

Here, H contains the multivariable nominal model P and themodel uncertainty structure. Specifically, P is recovered if� = 0, i.e., P = Fu(H , 0). In addition, the perturbation setis a norm-bounded subset of H∞

�u = {

�u ∈ RH∞∣

∣ ‖�u‖∞ ≤ γ}

(8)

where �u is of suitable dimensions in view of the LFT in (7).In principle, �u can be subject to additional structural con-straints, e.g., (block-) diagonality. However, as is argued inSection V, this leads to conservatism in the robust controllersynthesis step.

Associated with P is the worst-case performance criterion

JWC(P, C) = supP∈P

J (P, C).


Fig. 8. Model set P ={

P∣

∣P = Fu(H ,�u ),�u ∈ �u

}

.

By minimizing the worst-case performance

CRP = arg minC

JWC(P, C) (9)

it is guaranteed that

J (Po, CRP) ≤ JWC(P, CRP) (10)

hence CRP is guaranteed to result in a certain performancewhen implemented on the true wafer stage Po. In contrast,for a controller design that is solely based on the nominalmodel

CNP = arg minC

J (P, C) (11)

no such performance guarantees can be given, sinceJWC(P, CNP(P)) is arbitrary and may be unbounded. Thisis confirmed in Section V-B.

C. Toward a Joint System Identification and Robust ControlApproach

A key observation is that the guaranteed performance bound(10) highly depends on the model set P . To illustrate this,note that by making P large, it is generally easy to satisfythe constraint (6). However, the controller CRP has to achieveperformance with all candidate models in P , hence the worst-case performance bound in (10) is large and performance ispoor.

To ensure that P is small, the experimental conditionsshould be taken into account. As argued in Section II, thewafer stage system is open-loop unstable due to contactlessoperation. Hence, a stabilizing controller is needed. Thiscontroller is denoted Cexp. In this paper, the PID controllerwith a bandwidth of 40 Hz, as described in Section II-B3is used as Cexp. It is emphasized that Cexp is relativelystraightforward to tune using manual PID tuning, but does notmeet the performance requirements for normal wafer stageoperation. The crucial step in obtaining a small P lies inexploiting the knowledge of this controller that stabilized thesystem during the identification experiment.

The key idea in this paper is to identify the model set P suchthat it explicitly addresses the control objective JWC(P, C)and takes into account the experimental conditions. In partic-ular, the robust-control-relevant identification criterion

minP

JWC(P, Cexp)

subject to (6) (12)

Fig. 9. Identification and robust control design procedure.

is considered. Hence, besides connecting to the control cri-terion, the identification criterion (12) explicitly takes intoaccount the fact that the wafer stage system is stabilized byCexp during the identification experiment.

The main motivation for considering the robust-control-relevant identification criterion (12) is that this provides anupper bound for the resulting robust controller synthesis step,since it is directly verified that

JWC(P, CRP) ≤ JWC(P, Cexp).

D. Proposed Design Procedure

The resulting procedure is summarized in Fig. 9. Herein,Step 1 and Step 2 have already been performed in Sections II-A and III-A, respectively. In the next section, Step 3 and Step4 are performed that jointly lead to the model set P in (25).Then, Step 5, the robust controller synthesis and subsequentcontroller implementation are performed in Section V.

IV. ROBUST-CONTROL-RELEVANT IDENTIFICATION

In this section, the robust-control-relevant identificationproblem (12) is addressed. Hereto, the general LFT-baseduncertainty description (7) is used, which encompasses manyrelevant uncertainty structures in robust control theory. Theresulting model set depends on several choices:

1) the nominal model P that equals H22;2) the uncertainty structure that determines H11, H21, and

H12;3) the norm-bound γ in (8).

Since the goal is to solve (12), these choices are highlycoupled. To arrive at a tractable identification problem, the


identification of the nominal model P and the uncertaintymodeling step are separated.

In this section, the following steps are taken. First, inSection IV-A, preliminary steps are taken regarding the choiceof the uncertainty structure. This leads to a specific approachfor identifying a nominal model P in Section IV-B. Theinternal representation of the nominal model yields a newmodel uncertainty structure in Section IV-C that leads toa transparent separation of nominal model performance anduncertainty modeling. The application to the wafer stage ispresented in Section IV-D.

A. Model Uncertainty Structures in Identification for RobustControl

As is argued in the previous section, an uncertainty structureshould be selected that facilitates minimization of (12). Hereto,notice that the LFT-based description of P , see Fig. 8, can beadopted to express JWC(P, Cexp) as an LFT. This involvesthe construction of a generalized plant as in [9, Sec. 3.8].The uncertain model in Fig. 8 is appended with Cexp and theweighting functions, leading to Fig. 10. As a result

JWC(P, Cexp) = sup�u∈�u

∥

∥

∥

∥

Fu(M,�u)

∥

∥

∥

∥

∞

= sup�u∈�u

∥

∥

∥

∥

M22+M21�u(I −M11�u)−1 M12

∥

∥

∥

∥

∞.

(13)

The LFT (13) is a complicated function of �u . As a result, it isnot immediately clear how to actually minimize JWC(P, Cexp)over P . To simplify this, a model set P is constructed arounda nominal model P through a coprime-factor-based approach.Hereto, the nominal model P is internally structured as afactorization of two coprime factors, i.e., P = N D−1, where{N , D} should satisfy [22]:

1) N, D ∈ RH∞;2) ∃X, Y ∈ RH∞ such that X N + Y D = I

to be a right coprime factorization (RCF). Similarly, an RCF{Nc, Dc} of the controller Cexp is considered.

Next, consider the following dual-Youla-Kucera uncertaintystructure [25]–[27]:PDY =

{

P|P =(

N + Dc�u

) (

D − Nc�u

)−1, �u ∈ �u

}

.

(14)The model set (14) can also be cast in the LFT representation(7) by

H DY =[

D−1 Nc D−1

Dc + P Nc P

]

and direct computations reveal that the transfer function matrixM in Fig. 10 is given by the partitioned matrix

MDY(P, Cexp) =⎡

⎣

0 (D + Cexp N )−1 [

Cexp I]

V

W

[

Dc−Nc

]

W T ( P, Cexp)V

⎤

⎦ .

As a result, (13) reduces to an affine function in �u

JWC(PDY, Cexp) = sup�u∈�u

∥

∥

∥M22 + M21�u M12

∥

∥

∥∞ . (15)

Fig. 10. Worst-case performance JWC(P, Cexp) in generalized plant setting.

Hence, in contrast to the general LFT description in (13),JWC(PDY, Cexp) in (15) always is bounded.

The underlying mechanism is that the dual-Youla-Kucerauncertainty structure (14) parameterizes all candidate systemsthat are stabilized by Cexp. Besides the fact that this guaranteesthat JWC(PDY, Cexp) is bounded for any γ , it also ensures thatthe constraint (6) is satisfied for a certain γ .

Remark 1: Alternative uncertainty structures, including theadditive and multiplicative uncertainty structure, in generaldo not have the favorable properties associated with (14).These uncertainty structures lead to M11 �= 0, in which caseboundedness of JWC(P, Cexp) is not guaranteed. In addition,in that case also restrictive assumptions regarding the open-loop poles and zeros of P are required to satisfy (6), as isconfirmed in [12, Table 9.2].

Although the parameterization (14) leads to a boundedperformance JWC(PDY, Cexp), the actual minimization ofJWC(PDY, Cexp) over the model set PDY, see (12), is stillnot directly tractable. The key reason is that M21 and M12are frequency-dependent and multivariable transfer functionmatrices that influence the connection between the norm-bound γ in (8) and the worst-case performance criterionJWC(PDY, Cexp).

In the forthcoming sections, two steps are taken to solve(12). The key idea is to observe that the pairs {N , D}and {Nc, Dc} in (14) are any RCF of P and Cexp, respec-tively. Notice that infinitely many coprime factorizationsexist, i.e., coprime factorizations are not unique. In the nextSection IV-B, a new coprime factorization {N , D} of thenominal model P is defined. Then, in Section IV-C, itis shown that this coprime factorization leads to a newmodel uncertainty coordinate frame that enables a newsolution to (12).

B. Nominal Model Identification P

As is argued in Section IV-A, two steps are taken to identifythe model set in (12). First, a nominal model P is identified,which is the aim of this section. Second, the nominal model isextended with model uncertainty in Section IV-C. Both thesesteps are jointly aimed at solving (12).

1) Control-Relevant Identification: To ensure that the nom-inal model P is suitable to address (12), observe that theclosed-loop performance of any candidate model P is relatedto the true system through

J (Po, C) ≤ J (P, C) + ‖W (T (Po, C) − T (P, C)) V ‖∞ (16)

which directly results by application of the triangle inequalityto (3). By evaluating (16) for the controller Cexp and min-imizing over P , the following control-relevant identification


criterion is formulated:P = arg min

P

∥

∥W(

T (Po, Cexp) − T (P, Cexp))

V∥

∥∞ . (17)

The key idea behind the control-relevant identification cri-terion in (17) is that the nominal model is evaluated withrespect to closed-loop control objectives, which are specifiedby the control weighting filters W and V in (3). Note thatthe use of the triangle inequality in (16) is at the basis ofmany iterative identification and control techniques based onnominal models, including [28], and [29]. In this paper, thenominal model that minimizes the metric in (17) is shown to beespecially useful in identification for robust control based onmodel sets, see also (12). This will be shown in Section IV-C.The underlying technical result is a new connection between(17) and the identification of coprime factorizations, as isdiscussed in subsequent sections.

2) Coprime Factor Identification: As is discussed inSection IV-A, the nominal model P should be internallystructured as a coprime factorization to construct the modelset (14). These coprime factorizations are nonunique, since aninfinite amount of such factorizations exist. The key noveltyof this paper is the derivation of a new coprime factorizationthat uniquely connects control-relevant identification, see (17),and the identification of coprime factors. The key and uniqueadvantage of these specific coprime factorizations comparedto pre-existing coprime factorizations is that these facilitatedirect minimization in (12), as is shown in Section IV-C.

To proceed, let {Ne, De} be a left coprime factorization(LCF), see [22] for a definition, with co-inner numeratorof

[

CexpV2 V1]

, i.e., {Ne, De} is an LCF and satisfies theadditional condition that Ne N∗

e = I . Given Cexp, V2, and V1,such a coprime factorization can directly be computed, see [12]for details. Next, algebraic manipulations reveal that (17) isequivalent to

minN ,D

‖W

([

No

Do

]

−[

ND

])

Ne‖∞

subject to N , D ∈ RH∞ (18)

where[

ND

]

=[

PI

]

(

De + Ne,2V −12 P

)−1(19)

and Ne = [

Ne,2 Ne,1]

. In addition, the pairs {No, Do} and{N , D} are coprime factorizations of Po and P , respectively,as is proved in [30, Theorem 2]. It is emphasized that the pairs{No, Do} and {N , D} constitute a new robust-control-relevantfactorization, and are not equivalent to normalized coprimefactors, e.g., as used in [21] and [22].

The important aspect in (18) is that Ne is co-inner and doesnot influence the H∞-norm. Consequently, it can be removeddirectly, see (20), below. As a result, the four-block control-relevant identification problem (17) is recast as a two-blockcoprime factor identification problem.

3) Frequency Domain Algorithm: Solving the identificationproblem (18) is not immediate and several steps are requiredto arrive at a suitable identification algorithm. First, notice

that (18) involves an H∞ norm. By employing the frequency-domain interpretation of the H∞-norm, (18) is recast as

minN ,D

maxωi ∈�id

σ

(

W

([

No

Do

]

−[

ND

]))

subject to N , D ∈ RH∞. (20)

Second, {No, Do} is unknown. The key idea is thatT (Po, Cexp) can be directly identified using frequencyresponse estimation, see Appendix A for details, leading to anestimate ˜T (Po, Cexp) for ωi ∈ �id. A nonparametric estimateof {No, Do} is subsequently obtained by

[

˜No˜Do

]

= ˜T (Po, Cexp)V N∗e for ωi ∈ �id. (21)

Finally, it remains to determine the optimal model {N , D} in(18). Hereto, the model is parameterized as

[

N (θ)

D(θ)

]

=[

B(θ)A(θ)

]

(De A(θ) + Ne,2V −12 B(θ))−1. (22)

This parameterization exploits knowledge of Cexp and effec-tively connects stability of the factors {N , D} and closed-loop stability of the model, see [30, Th. 4] for a proof. Inaddition, the dynamics that are introduced by the experimentalcontroller Cexp and weighting filters V2 and V1 in (22) cancelout exactly when constructing P , since

P(θ) = N (θ)D(θ)−1 = B(θ)A(θ)−1. (23)

Here, B, A ∈ R2×2[z], i.e., polynomial 2 × 2 matrices in the

complex indeterminate z, see [31, Sec. 7.21] for more details.Hence, P(θ) in (23) is parameterized as a matrix fractiondescription (MFD). By using a so-called full polynomial form,see [32, Ch. 6], the common dynamics between differentinput–output channels are taken into account. This leads tomodels with a low McMillan degree. Due to the one-to-one correspondence between MFDs and state-space models,this directly leads to state-space models with a small statedimension.

The actual minimization of (20) is performed using theLawson algorithm in [33]. This algorithm is specificallytailored toward complex multivariable motion systems toprovide a numerically optimal conditioning during theiterative algorithm.

C. Robust-Control-Relevant Model Set PIn the previous section, a new coprime factorization of P

has been introduced that can be directly identified from datain a control-relevant manner. In this section, the model uncer-tainty coordinate frame is further defined, such that P andthe model uncertainty jointly aim at solving (12). Hereto, thespecific robust-control-relevant coprime factorization {N , D}of P , see Section IV-B is considered in conjunction with a new(Wu, Wy)-normalized RCF of Cexp. Here, the pair {Nc, Dc} isa (Wu, Wy)-normalized RCF of Cexp if it is an RCF of Cexp

and, in addition[

Wu Nc

Wy Dc

]∗ [

Wu Nc

Wy Dc

]

= I. (24)


The computation of such a (Wu, Wy)-normalized RCF ofCexp follows directly from the solution of a Riccati equation,see [30] for details. Next, let the model set PRCR be definedas

PRCR ={

P|P ∈ PDY, {N , D} satisfies (19), {Nc, Dc} satisfies (24)}

.

(25)

Essentially, PRCR in (25) is constructed as in (14) with aspecific choice of coprime factorizations {N , D} and {Nc, Dc}of P and Cexp, respectively.

The main result associated with PRCR is the result

JWC(PRCR, Cexp)

= sup�u∈�u

‖MRCR22 + MRCR

21 �u MRCR12 ‖∞ (26)

≤ ‖MRCR22 ‖∞ + sup

�u∈�u

‖MRCR21 �u MRCR

12 ‖∞ (27)

= J (P, Cexp) + γ. (28)

Here, (27) follows by application of the triangle inequality.Moreover, (28) follows from the observation that ‖MRCR

22 ‖∞ =J (P, Cexp) and MRCR

21 and MRCR12 are norm-preserving by the

specific coprime factorizations of Cexp and P , respectively.See [30, Theorem 9] for a detailed proof of (28).

The key point in regarding the model set PRCR in (28) isthat the size of model uncertainty γ directly relates to theworst-case performance criterion JWC(P, Cexp) in (12). As aresult, the uncertainty structure (25) reduces solving (12) todetermining the value of γ such that (6) holds.

To corroborate the statement that the separate nominalmodel identification in Section IV-B and uncertainty modelingprocedure in this section jointly minimize (12), notice that(17) is minimized during nominal model identification. Thisterm exactly corresponds to γ in (28). Hence, the nomi-nal modeling procedure in Section IV-B and an uncertaintymodeling approach using the structure (25) jointly aim atminimizing (12).

Remark 2: In the case where the uncertainty structure (14)is adopted, but different coprime factorizations are used thanthe ones for PRCR in (25), the resulting worst-case per-formance bound (15) is obtained. This holds for instancein the situation where commonly used normalized coprimefactorizations, see [21], [22], are adopted. In such a situation,M21 and M12 are frequency-dependent multivariable transferfunction matrices. As a result, a finite γ can lead to anarbitrarily large, yet bounded, JWC(PDY, Cexp). This has thefollowing important implications.

1) If unstructured model uncertainty is used in conjunctionwith the model set PDY, i.e., using arbitrary coprimefactorizations, then the resulting JWC(PDY, Cexp) maybe arbitrarily large, leading to conservatism in the iden-tification step.

2) The effects of M21 and M12 can be mitigated by consid-ering highly structured perturbations, as is proposed in[7, Sec. 2.4] and also applied in [16]. Such a procedurereduces conservatism in the identification step. However,it leads to a highly structured perturbation model consist-ing of ny × nu scalar perturbation blocks. As a result,

such a procedure leads to conservatism in the robustcontroller synthesis (9), as is shown in Section V.

D. Wafer Stage Nominal Model Identification: Step 3 in Fig. 9

In view of Step 3 in Fig. 9, a nominal parametric model isidentified that is internally structured as a coprime factoriza-tion {N , D}. Hereto, the weighting functions in Section III-A2are employed together with a frequency response functionestimate ˜T (Po, Cexp) that is obtained using the procedure inAppendix A. Next, the coprime factor frequency responsefunction {˜No, ˜Do} is computed for ωi ∈ �id using (21). Theresults are depicted in Fig. 11. Due to the dedicated multisineexperiment design in conjunction with a high signal-to-noiseratio, the variance error is negligible and consequently notshown in Fig. 11.

Subsequently, the identification problem (20) is solved usingthe algorithm in [33]. The model order is selected using theresults in [34]. The McMillan degree of the optimal modelequals 8.

The resulting identified coprime factors {N , D} are depictedin Fig. 11. In addition, the open-loop frequency responsefunction Po and model P = N D−1 = B A−1 are comparedin Fig. 12, which facilitates the interpretation of the modelin terms of physical system properties. Note that due to thespecific parameterization (23) in terms of a matrix fractiondescription, a minimal state space model of P has statedimension 8.

Analysis of the identified model in Fig. 12 that has minimalstate dimension 8 reveals that it is of low order for two reasons.First, the model only represents a limited number of resonancephenomena of ˜Po that are observed in Fig. 12. This is a directconsequence of the control-relevant identification criterion in(17). This can also be observed from the identified coprimefactors in Fig. 11 that directly connect to control-relevance interms of (17). In particular, system dynamics that have a highgain in the coprime factor domain are important for controland should be modeled accurately, which is clearly the casein Fig. 11. Finally, these results in Fig. 12 confirm controldesign experience for the wafer stage system, since the firstresonance phenomena indeed limit the performance of typicalPID control designs, see also Section II-B3.

Second, the identification procedure directly identifies amultivariable model that takes into account common dynamicsbetween the different inputs and outputs. This is enabled bythe multivariable parameterization in (23). In the multivariablemodel, four states correspond to the two rigid-body modesin both the x-direction and the y-direction. The other fourstates correspond to two resonance phenomena at 208 and214 Hz, which each consists of a complex pair of eigenvalues.Interestingly, these resonance phenomena correspond to mul-tivariable system behavior, since both these resonances appearin all four transfer functions in Fig. 12, yet only require twostates each.

E. Wafer Stage Uncertainty Modeling: Step 4 in Fig. 9

The identified coprime factorization in Section IV-D is usedto construct the robust-control-relevant model set PRCR in


Fig. 11. Coprime factorization: identified frequency response functionNo , Do for ωi ∈ � (solid blue, dotted), eighth-order parametric modelcoprime factorizations N , D (dashed red).

(25). As a result, the important result (28) applies

JWC(PRCR, Cexp) ≤ J (P, Cexp) + γ.

The nominal model P = N D−1 is already identified andleads to J (P, Cexp) = 18.17. Hence, it now remains toestimate γ such that (6) holds. The estimation of γ usinga validation-based uncertainty modeling approach is the aimof this section.

1) Validation-Based Uncertainty Modeling Approach: Theaim of this section is to determine γ by using the modelvalidation procedure in [35], which is an extension of theapproach in [36]. Herein, the model quality is tested using boththe identification data set and many different independent data

Fig. 12. Bode magnitude diagram: nonparametric estimate (dot), nominalmodel P (solid blue), model set Pdyn (yellow shaded), and Psta (cyanshaded).

sets that are collected under closed-loop operating conditionswith Cexp implemented on the wafer stage system Po.

The result of the considered validation-based uncertaintymodeling procedure is the minimum-norm bound

γ (ωi ) = σ (�u(ωi )), for ωi ∈ �id ∪ �val (29)

where �u is unstructured as in (8) and �val contains thefrequencies that are present in the validation data sets.

The design procedure in Fig. 9 requires validation data toconstruct the model set PRCR. Hereto, data are collected underseveral relevant operating conditions:

1) data sets are collected under identical conditions as inSection IV-B3, i.e., containing frequency componentsωi ∈ �id and identical phases φk as in (35);

2) data sets are collected with different frequency compo-nents than those used in Section IV-B3, i.e., the inputsignal w in this case contains frequencies ωi ∈ �val,where ωi /∈ �id;

3) data sets are collected where the input contains frequen-cies ωi ∈ �id, but with random phases φk in (35).

The norm-bound γ (ωi ), see (29), that results from thedifferent validation experiments is depicted in Fig. 13. Therationale behind the data sets associated with Item 2) withfrequency components that are not contained in �id is toinvestigate possible interpolation errors due to the use of adiscrete frequency grid in the approximation (20) to (18). Fromthe model validation results in Fig. 13, it can be seen thatthe model error on the validation frequency grid �val is notsignificantly larger in comparison to �id.

The purpose of varying the phases φk in Item 3) is toinvestigate the presence of parasitic nonlinear effects, whichis closely related to the approach suggested in [31, Ch. 3].It appears that the use of different phases results in aslightly larger value γ (ωi ), especially around the resonancephenomena, see also ˜Po(ωi ) in Fig. 7 for the correspondingfrequencies. Similar results regarding the identification offlexible dynamical behavior have been reported in [11].


Fig. 13. Resulting γ (ωi ) from (29): data sets: 1) on identification grid(blue ×); 2) on validation grid (green �); and 3) with varying phases φk(red ◦), and parametric overbound resulting in Pdyn (solid blue).

TABLE I

ROBUST-CONTROL-RELEVANT IDENTIFICATION AND ROBUST

CONTROLLER SYNTHESIS RESULTS. HEREIN, ∞ IMPLIES THAT

THE CRITERION IS UNBOUNDED. SEE ALSO SECTION III-A2

FOR A FURTHER EXPLANATION

Minimized fBW JWC JWCController Criterion J ( P, C) ( P, C) (Psta, C) (Pdyn, C)

Cexp None (PID) 18.17 40.4 18.60 18.21CNP J ( P, C) 2.21 83.9 ∞ ∞CRP JWC(Pdyn, C) 3.31 69.1 ∞ 3.32

Next, the static model uncertainty bound γ in (8) can beobtained from

γ = supωi ∈�id∪�val

γ (ωi ) = 0.43

leading to the model set

Psta ={

P ∈ PRCR|‖�u‖∞ ≤ 0.43}

where �u are unstructured perturbations. Since the model setPsta is not invalidated by the many data sets, the true systemis expected to satisfy Po ∈ Psta.

When investigating the worst-case performance associatedwith Psta, i.e., JWC(Psta, Cexp) = 18.60, see Table I, then itis clear that the bound (26) holds and is tight in this case,since J (P, Cexp) = 18.17. The model set Psta is thus robust-control-relevant in the sense of (12).

2) Analysis of the Robust-Control-Relevant Model Set Psta:To illustrate the properties of the identified robust-control-relevant model sets, the visualization procedure that is devel-oped in [37] is invoked. The resulting Bode diagrams interms of the elementwise amplitude and singular values aredepicted in Figs. 12 and 14. It is emphasized that these Bodediagrams correspond to open-loop system dynamics. Hence,these enable an analysis of physical system properties that areaccurately modeled in the model set P .

The Bode diagrams in Figs. 12 and 14 reveal that the modelis most accurate in the cross-over region around 90 Hz, wherethe rigid-bode behavior and the two resonance phenomenaare very accurately modeled. Interestingly, at low frequencies,where the rigid-body mode dominates the system behavior, themodel set is relatively large and hence uncertain. A similar

Fig. 14. Elementwise Bode magnitude diagram: nominal model P (solidblue), model set Pdyn (yellow shaded), and Psta (cyan shaded).

observation holds at higher frequencies, where the uncertaintyassociated with model set increases for increasing frequencies.

The particular shape of the model set Psta is attributedto the specific choice of coprime factorization in (25).It is emphasized that the corresponding norm-bound γ in (8)is unstructured, hence it is constant for all frequencies andinput/output directions.

3) Further Refinements Through the Use of a DynamicUpper Bound: Since γ (ωi ), ωi ∈ �id ∪ �val is frequency-dependent, it can also be overbounded by a dynamic weightingfunction Wγ , with Wγ , W−1

γ ∈ RH[1×1]∞ . The use of Wγ leadsto the model set

Pdyn ={

P ∈ PRCR|‖�u W−1γ ‖∞ ≤ 1

}

(30)

where it can be shown by means of a Nevanlinna–Pickinterpolation argument that Po ∈ Pdyn if γ (ωi ) ≤ Wγ forωi ∈ �id ∪ �val, see [35] for a proof. If the overbound Wγ ischosen tight, i.e., it satisfies ‖Wγ ‖∞ = supωi∈�id∪�val γ (ωi ),then the use of the dynamic upper bound Wγ does not affectthe bound in (28). However, there are at least three reasonsfor introducing a dynamic overbound Wγ in the model set:

1) since Pdyn ⊆ Psta, it always results in a nonincreasingworst-case performance compared to a static overboundas in Psta, hence it can only reduce possible conser-vatism;

2) it can reduce potential conservatism that is introducedby the approximation in Section III-C, where Cexp isused instead of the optimal choice CRP(P), see [34] fora detailed argumentation;

3) it can reduce conservatism that is introduced by the useof the upper bound in the triangle inequality in (26), seealso [34].

The overbound Wγ in (30) that is used in this paper isdepicted in Fig. 13. When comparing the resulting modelset Pdyn in (30) with Psta, then it is observed from Table Ithat the worst-case performance associated with Pdyn equalsJWC(Pdyn, Cexp) = 18.21. This is a slight reduction compared


to Psta, since JWC(Psta, Cexp) = 18.60. This small improve-ment is attributed to the use of the upper bounds in the triangleinequality that is used in the result (28).

A comparison of Psta and Pdyn in Figs. 14 and 12, confirmsthat Pdyn ⊂ Psta. Hence, the use of a dynamic overboundin (30) leads to a reduction of possible conservatism. It isemphasized that both model sets Psta and Pdyn are constructedusing the unstructured perturbation model (8).

V. ROBUST CONTROLLER SYNTHESIS

A. Theory

In this section, the identified robust-control-relevant modelset Pdyn in Section IV is used as a basis for robust controllersynthesis in (9), where the performance objectives have beenspecified in Section III-A. The optimal robust controller is thenimplemented on the true system.

First, the actual robust controller synthesis (9) is recast interms of a structured singular value synthesis. Hereto, consider

G =⎡

⎢

⎣

Wγ D−1 Nc 0 Wγ D−1V1 Wγ D−1

Wy(Dc + P Nc) 0 Wy PV1 Wy P0 0 Wu V1 Wu

−(Dc + P Nc) V2 −PV1 −P

⎤

⎥

⎦

hence M = Fl(G, Cexp), see also (13). Then

CRP = arg minC

JWC(P, C)

= arg minC stabilizing

supω∈[0,2π)

μs(

Fl

(

G(

e jω)

, C(

e jω)))

(31)

whereμs (.) = max

∥

∥

∥

∥

[

qw

]∥

∥

∥

∥

2=1

{

α

∣

∣

∣

∣

‖q‖2 ≤ γ ‖p‖2α‖w‖2 ≤ ‖z‖2

}

(32)

q = �u p, and z = Fu(M,�u)w.In (32), the structured singular value arises due to the

fact that the performance channel w �→ z and uncertaintychannel q �→ p are separated. The structured singular value isskewed due to the fact that the norm-bound on the uncertaintychannel q �→ p is fixed to γ , while the norm-bound on theperformance channel w �→ z is to be minimized.

An efficient approach to solve the skewed structured sin-gular value problem (32) is through D − K -iterations. InD − K -iterations, a μ-analysis (D-step) and H∞-optimization(K -step) are solved alternately. The key advantage of usingthe unstructured perturbation model (8) is that the channelsq �→ p do not contain any further structure. This hasas important property that the μ-analysis step (D-step) canbe solved without conservatism. This property is known asμ-simple [17, Sec. 9]. In contrast, if the model perturbationin (8) contains additional structure, then the resulting skewedstructured singular value problem is not μ-simple, in whichcase the robust controller CRP is unnecessarily conservative.The underlying reason is the fact that the upper bounds that areused in μ-analysis are tight only if the uncertainty structure isμ-simple.

Remark 3: Although the iterative D − K -iteration involvesa sequence of convex optimization problems, the robust con-troller synthesis problem is non-convex and need not converge

Fig. 15. Bode diagram: initial controller Cexp (solid blue), CRP (dashedred), and CNP (dash-dotted green).

to an optimum. However, as is also claimed in [9, Sec. 8.12.1],the algorithm performs well in practice.

B. Robust Controller Synthesis and Implementation: Step 5 inFig. 9

The identified model set Pdyn is used to synthesize arobust controller using (31). For comparison, also a nominalcontroller CNP, see (11), is synthesized using P and standardH∞-optimization [12]. The resulting controllers are depictedin Fig. 15, whereas the resulting performance of the controllerswhen evaluated on the model is given in Table I. Severalobservations are made. First, the controller CNP achieves opti-mal performance for the nominal model P , i.e., J (P, CNP)= 2.21. However, the worst-case performance associated withPdyn is unbounded, hence neither stability nor performancecan be guaranteed when implementing CNP on the true systemPo. Clearly, this underlines the necessity of a robust controldesign for high performance motion control.

Second, the controller CRP achieves optimal worst-caseperformance for the model set Pdyn. The bounds

JWC(Pdyn, CRP) ≤ JWC(Pdyn, Cexp)

JWC(Pdyn, CRP) ≤ JWC(Pdyn, CNP)

hold as is expected. In addition, the worst-case performancecompared to Cexp is significantly reduced. Hence, CRP leadsto a significantly improved guaranteed performance whencompared to Cexp. Note that this performance is measuredin terms of the performance criterion. If the criterion is small,then this implies that the achieved loop-shape better matchesthe desired loop-shape in Fig. 7, see Section III-A for anexplanation. This corresponds to the achieved closed-loopbandwidth fBW in Table I. Here, closed-loop bandwidth ofthe multivariable system refers to the smallest bandwidth forthe x-direction and y-direction, see also Section II-B1. Indeed,note that the control goal was set to a desired bandwidth


Fig. 16. Nyquist diagram of characteristic loci λ(CRP P). Evaluated forrobust optimal controller CRP using P (solid blue) and identified frequencyresponse function ˜Po(ωi ) (dotted).

of 90 Hz, see Section III-A2. The controller CNP almostachieves the desired bandwidth. Although the controller CRP

leads to a significant increase of fBW when compared to Cexp,the need for robustness against modeling errors leads to alower bandwidth when compared to CNP. Although this isessential to guarantee closed-loop stability when implementingthe resulting controller on the true system, the bandwidth of69.1 Hz implies that disturbances beyond 70 Hz will not beattenuated.

Third, in contrast to the initial controller Cexp, the optimalcontrollers CNP and CRP are inherently multivariable. Hereto,notice that the inputs and outputs of the wafer stage systemhave equal units and comparable magnitude. Next, in Fig. 15it is observed that the off-diagonal elements have a magnitudecomparable to the diagonal elements in the high-frequencyrange. Clearly, the compensation of the multivariable reso-nance phenomena benefits from a multivariable controller.

Fourth, to further interpret the behavior of the optimal robustcontroller, the characteristic loci of the loop-gain λ(CRP P),see [9, Sec. 4.9.3], are depicted in Fig. 16, both evaluatedon the nominal model P and identified frequency responsefunction ˜Po(ωi ), ωi ∈ �id. Interestingly, around the resonancephenomena, the loop-gain has an amplitude that is signifi-cantly larger than one. However, the corresponding phase isbetween approximately −90 and 90 degrees. Hence, in view ofclosed-loop stability of a multivariable Nyquist criterion, theloop-gain does not encircle the point −1 in the complex plane.The diagonal entries of the closed-loop sensitivity function(I +CRP P)−1 as depicted in Fig. 17 corroborate these results,since their magnitude drops below 1 (0 dB) around severalresonance phenomena. Moreover, Fig. 17 reveals that thesensitivity functions evaluated on the nominal model P , i.e.,(I+C P)−1, and evaluated on the identified frequency response

Fig. 17. Bode magnitude diagram of diagonal elements of closed-loopsensitivity function (I + C P)−1: evaluated on nominal model P (solid,colors corresponding to Fig. 15) and identified frequency response function˜Po(ωi ), ωi ∈ �id (red dots and dashed lines). Cexp (top), CNP (middle), andCRP (bottom).

function, i.e., (I + C ˜Po)−1 for ωi ∈ �id closely overlap for

Cexp and CRP. However, the transfer function (I + CNP˜Po)

−1

has significant peaks of approximately 40 dB, which confirmsthe poor robustness properties associated with CNP. Finally,the closed-loop sensitivity functions in Fig. 17 confirm theincreased bandwidth when comparing Cexp and CRP andimproved low-frequency disturbance attenuation properties.

Next, the controllers Cexp and CRP are implemented onthe true wafer stage system Po. The controller CNP is notimplemented, since it destabilizes the system, see also Table I.In contrast, since the model set P is assumed to encompass thetrue wafer stage system Po, see (6), and Cexp and CRP stabilizeall candidate models in P , these controllers are guaranteed tostabilize the wafer stage system.

The evaluation of standstill errors is an important perfor-mance indicator for wafer stage systems. Here, the referencesignal is set to zero, leading to a regulator problem withperformance variable e = −y. In this case, the key task ofthe feedback controller is to attenuate exogenous disturbancesthat affect the wafer stage.

The resulting time-domain measurements are depicted inFig. 18, whereas the cumulative power spectrum (CPS) isdepicted in Fig. 19. In addition, the standard deviation


Fig. 18. Measured error signals in x-direction (left) and y-direction (right):initial controller Cexp (top), CRP (bottom).

Fig. 19. Cumulative power spectrum of the measured error signal in x-direction (left) and y-direction (right): initial controller Cexp (solid blue),CRP (dashed red).

TABLE II

STANDARD DEVIATION AND PEAK ERROR SIGNALS IN [nm]

σ x σ y px py

Cexp 25.3 13.5 129.4 60.4CRP 12.7 6.4 54.1 27.4

σ x and σ y and peak values px and py for the x-directionand y-direction, respectively, are given in Table II. First, itis observed from Fig. 19 that the disturbances with controllerCexp implemented are mainly present in the lower frequencyrange. Hence, it is expected that increasing the bandwidthof the controller and hence reducing the sensitivity functionat lower frequencies (Fig. 17) leads to improved disturbanceattenuation properties. This is best visible by analyzing thespectra of the measured error signals in Fig. 19. As a result,also the variance of the error reduces by approximately afactor two when comparing Cexp and CRP, see Table II.In conclusion, the controller CRP significantly improves themeasured performance.

Finally, it is observed from Figs. 18, 19, and Table II thatthe measured errors in the x-direction are significantly largerthan the errors in the y-direction for both controllers Cexp andCRP. This is explained by the fact that a cable-arm is present

in the x-direction that is used to provide electrical current andwater cooling to the actuation system, see also Section II-A.

VI. CONCLUSION

In this paper, a novel framework for next-generation motioncontrol was presented and implemented on an industrialwafer stage system. The framework was tailored toward next-generation motion systems that are expected to be increasinglycomplex in the sense of high order flexible dynamics andan increasing number of inputs and outputs. The frameworkencompasses all steps from system identification and robustcontrol to controller implementation. Experimental results ofthe control implementation on an industrial wafer stage systemconfirmed a significant performance improvement.

The resulting framework is particularly suitable for next-generation motion systems with many inputs and outputs, sincethe complexity of the uncertainty model does not inflate whenthe number of inputs and outputs increases. This was achievedby a multivariable model parameterization that takes intoaccount common dynamics between various inputs and outputsin conjunction with the nonconservative use of unstructuredperturbation models. The key technical result that leads to thenonconservative use of these unstructured perturbation modelsinvolves new coprime factorization results. Subsequently, theuse of such unstructured model uncertainty descriptions hasimportant advantages for robust controller synthesis, sinceexisting μ-synthesis techniques lead to a nonconservative con-troller design for the proposed model set. This is in sharp con-trast to pre-existing approaches, where the use of highly struc-tured uncertainty models leads to conservatism in μ-synthesis.

Continued research focuses on the following aspects.1) The limits of achievable control performance for the

traditional motion control situation are being investigated,where the number of inputs and outputs equals the num-ber of motion DOFs. Subsequently, potential performanceimprovement can be achieved when additional actuators andsensors are available, see also Section I-C. An additionalaspect involves the placement of actuators and sensors in viewof the achievable control performance.

2) The control goal definition, which in this paper is entirelybased on loop-shaping techniques (Section III-A2), should besystematically extended. These loop-shaping techniques andthe derived bandwidth specification are based on approxi-mate knowledge of the exogenous disturbances that affectthe true system. For improved disturbance attenuation andthus improved control performance, the control goal shouldbe extended toward explicitly incorporating models of thedisturbances that affect the true system, see Fig. 19. Initialresults in this direction include the ad hoc procedure in [13,Sec. 4.3].

3) Unmeasured performance variables, see Section I-C,should be appropriately dealt with. Although conceptually thisfits in the standard plant formulation [9, Sec. 3.8], significantextensions of the procedure that is presented in this paper arerequired. Initial results in this direction are presented in [38].

4) Possible position-dependency of the wafer stage dynam-ics should be investigated. Obviously, motion systems involve


moving parts of the system. Consequently, the dynamics of thesystem generally depend on the operating conditions. Althoughan initial attempt to improve the control performance throughposition-dependent modeling of a similar system is presentedin [39], see also [40] for a related approach, the resultingposition-dependent controller does not significantly improvethe performance. A possible explanation for the lack of per-formance improvement is the fact that the position-dependentdynamics are not important from a control perspective, whichis experimentally investigated in [41, Sec. 5.7] using theresults of this paper. These observations imply a need for alsoincorporating control-relevance, as is done in this paper, inthese position-dependent identification techniques.

APPENDIX

A. Frequency Response Function Identification

In this section, frequency response function identification isinvestigated. For the sake of exposition, attention is restrictedto dim u = dim y = 2, as in Section II-A.

To identify frequency response functions, both the excitationsignal r1 and the signals u, and y are measured in the setupof Fig. 5. These signals are transformed into the frequencydomain through the Fourier transform. This leads to R< j>

1 ∈C

2×1, U< j> ∈ C2×1, and Y< j> ∈ C

2×1 for ωi ∈ �, where �denotes the standard DFT grid and the argument ωi is omittedfor notational reasons. Furthermore, the superscript < j> refersto experiment j . By performing two experiments, i.e., j =1, 2, it follows from (5) that

[Y<1> Y<2>

U<1> U<2>

]

=[

PoI

]

(I + Cexp Po)−1 [R<1>1 R<2>

1

]

. (33)

Then, an estimate of T (Po, Cexp) is given by

˜T (Po, Cexp) =[Y<1> Y<2>

U<1> U<2>

]

[R<1>1 R<2>

1

]−1 [

Cexp I]

(34)

for ωi ∈ �id, �id = {

ω∣

∣ω ∈ �, det([

R<1>1 R<2>

1

]) �= 0}

.Several aspects are important with respect to the experimentdesign. First, under the mild assumption that v in Fig. 5 isfiltered white noise, U< j>(ωi ) and Y< j>(ωi ) are circularly com-plex normally distributed with zero mean. Consequently, theestimation error introduced by the noise v can be representedsolely by the variance. Periodic input signals are employedto reduce the variance of the estimate in (34) for increasingmeasurement length. Notice that the key advantage of usingsuch periodic input signals is that the use of windowing isrendered superfluous, hence no bias errors are introduced inthe estimation step. Such multisine input signals are alsohighly recommended in [31, Sec. 2.8.1]. Specifically

[

r<1>1 (t) r<2>

1 (t)] = Q

∑

k

ak sin(ωk t + φk) (35)

where ωk ∈ �id, ak is the corresponding amplitude and φk

the phase. A necessary condition for det([R<1>

1 R<2>1

]

) �= 0is that Q is of full rank. This means that an excitation hasto be applied in both the x-direction and the y-direction suchthat the entire input space is spanned. In this paper, this isachieved by sequential excitation in the x and y direction indifferent experiments, leading to Q = I .

To obtain a frequency response function estimate of Po,observe that T (P, C) in (5) can be partitioned as

T (P, C) =[

T11 T12T21 T22

]

=[

PI

]

(I + C P)−1 [

C I]

(36)

hence P = T12T −122 . As a result, given ˜T (Po, Cexp) for ωi ∈

�id, then for each ωi ∈ �id, ˜Po = ˜T12(

˜T22)−1

.

ACKNOWLEDGMENT

The authors would like to thank their colleagues at PhilipsApplied Technologies Eindhoven, The Netherlands, for pro-viding the experimental facilities and for their fruitful discus-sions. In addition, the Associate Editor and the Reviewers aregratefully acknowledged for their constructive comments.

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[41] T. Oomen, “System identification for robust and inferential controlwith applications to ILC and precision motion systems,” Ph.D. dis-sertation, Eindhoven Univ. of Technol., Eindhoven, The Netherlands,2010.

Tom Oomen received the M.Sc. degree (cum laude)and the Ph.D. degree from the Eindhoven Universityof Technology, Eindhoven, The Netherlands, in 2005and 2010, respectively.

He is currently an Assistant Professor withthe Department of Mechanical Engineering, Eind-hoven University of Technology. He was a VisitingResearcher with the KTH, Stockholm, Sweden, andwith The University of Newcastle, Newcastle, Aus-tralia. His current research interests include identifi-cation and robust control, with applications in high-

precision mechanical servo systems.Dr. Oomen was a recipient of the Corus Young Talent Graduation Award

in 2005.

Robbert van Herpen received the M.Sc. degree inelectrical engineering (cum laude) from the Eind-hoven University of Technology, Eindhoven, TheNetherlands, in 2009, where he is currently pursuingthe Ph.D. degree in system identification for robustcontrol of motion systems that are controlled usinga large number of actuators and sensors with theMechanical Engineering Department.

Sander Quist received the M.Sc. degree (cum laude)in electrical engineering from the Eindhoven Univer-sity of Technology, Eindhoven, The Netherlands, in2010, where his Master’s thesis on robust control fora flexible wafer stage was done with Philips AppliedTechnologies.

He joined Cargill, The Netherlands, in 2010, as aTechnical Management Trainee.

Marc van de Wal received the Ir. degree in mechan-ical engineering and the Ph.D. degree in controlstructure design, including actuator and sensor selec-tion from the Eindhoven University of Technology,Eindhoven, The Netherlands, in 1993 and 1998,respectively.

In 1998, he joined Philips, Eindhoven, wherehe is currently involved in research on advancedmechatronics and motion control for high-precisionelectromechanical servo systems, mainly wafer scan-ners for the semiconductor industry. His current

research interests include robust control, gain-scheduling and LPV control,and advanced feedforward control methods.

Okko Bosgra received the M.S. degree andResearch Diploma from the Delft University ofTechnology, Delft, The Netherlands.

He was a Professor of systems and control withWageningen University, from 1980 to 1985. Since1986, he has been the Chair of the MechanicalEngineering Systems and Control Group, Delft Uni-versity of Technology. Since 2003, he has beenwith the Eindhoven University of Technology, Eind-hoven, The Netherlands. His current research inter-ests include applications of robust control and sys-

tem identification in process control and motion control.

Maarten Steinbuch received the M.Sc. and Ph.D.degrees from the Delft University of Technology,Delft, The Netherlands.

He is currently a Professor of systems and controland the Head of the Control Systems TechnologyGroup, Mechanical Engineering Department, Eind-hoven University of Technology, Eindhoven, TheNetherlands. He was with Philips, Eindhoven, from1987 to 1999. His current research interests includemodeling, design, and control of motion systems,robotics, automotive powertrains, and control of

fusion plasmas.Dr. Steinbuch is the Editor-in-Chief of IFAC Mechatronics, and an Associate

Editor of the International Journal of Powertrains. Since 2006, he has beenthe Scientific Director of the Center of Competence High Tech Systems ofthe Federation of Dutch Technical Universities.

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