Exploiting Additional Actuators and Sensors for Nano-Positioning Robust Motion Control

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1 204 American Control Conference (ACC) June 4-6, 204. Portland, Oregon, USA Exploiting Additional Actuators and Sensors for Nano-Positioning Robust Motion Control Robbert van Herpen, Tom Oomen, Edward Kikken, Marc van de Wal, Wouter Aangenent, Maarten Steinbuch Abstract The ongoing need for miniaturiation and an increase of throughput in IC-manufacturing is obstructed by performance limitations in motion control of nano-positioning wafer stages. These limitations are imposed by flexible dynamical behavior, associated with structural deformations of the nano-positioning stages. The aim of this research is to investigate limits on achievable performance in a conventional control configuration and to mitigate these limits through the use of additional actuators and sensors. To this end, a systematic framework for control design using additional actuators and sensors in the generalied plant configuration is presented, which leads to a well-posed H -control optimiation problem that extends conventional design approaches in a natural way and exploits physical insight to address structural deformations in weighting filter design. Through an experimental confrontation of the design framework with a prototype next-generation nano-positioning motion system, successful performance enhancement beyond the conventional limits is demonstrated. I. INTRODUCTION Increasing demands for high machine throughput of nanopositioning systems necessitates explicit control of flexible dynamical behavior. An important application domain where positioning with nanometer accuracy plays a central role is the lithography step in IC-manufacturing, see, e.g., []. Herein, a desired IC-pattern is etched into a photosensitive layer on a substrate, using a device known as a wafer scanner. For this purpose, the substrate, a silicon wafer, is placed on top of a wafer stage, see Fig., to position it with respect to the light source. On the one hand, a nanometer positioning accuracy has to be achieved in view of the required feature sie of current ICs. On the other hand, it is vital for market viability of wafer scanners to achieve a high machine throughput. Since a high throughput demands for large accelerations of the wafer stage, next-generation stages are designed to be lightweight, as is further motivated in [2]. However, lightweight stages tend to undergo structural deformations upon large accelerations, which obstruct the intended positioning accuracy. Therefore, it is essential to explicitly control these structural deformations. Flexible dynamical behavior of lightweight positioning stages leads to limitations on the achievable motion performance. Fundamental performance limitations for feedback control are formulated in terms of sensitivity integrals, which depend on right-half plane poles and eros of the system, [3], [4]. Extensions of these classical results towards the generalied plant framework for model-based control are provided in, e.g., [5], [6]. In turn, the generalied plant framework Van Herpen, Oomen, Kikken, and Steinbuch are with the Eindhoven University of Technology, Department of Mechanical Engineering, Control Systems Technology Group, Eindhoven, The Netherlands ( R.M.A.v.Herpen@tue.nl). Van de Wal and Aangenent are with ASML Research, Veldhoven, The Netherlands. Fig.. Wafer stage Metrology frame A prototype lightweight wafer stage, enabling high accelerations. encompasses two degree-of-freedom control configurations, as for example used in inferential control to account for unmeasured performance variables [7]. Although fundamental performance limitations are wellunderstood, the standard formulation does not immediately indicate that performance limitations in motion control result from structural deformations. For example, flexible dynamical behavior does not necessarily induce right-half plane poles and eros, to which bandwidth limitations have been associated, see [8, Sect. 5.7, 5.9]. Nevertheless, it is known from practical experience that the attainable bandwidths in motion control are dominantly restricted by flexible dynamical behavior, as is confirmed by the results in [9], [0], []. To formalie this limitation, it is essential to explicitly involve model uncertainty in a robust control setting, as is also argued in [2] and [8, Sect ]. The main contribution of this paper is a systematic robust control design approach to beat conventional limits through active control of flexible dynamical behavior using additional actuators and sensors. Based on physical insight, new weighting filters for H control design are proposed. The design framework is confronted with an industrial high-precision motion system, demonstrating an enhancement of motion performance beyond conventional bandwidth limitations. The paper is organied as follows. In Sect. II, a generalied plant framework is adopted. To address these limits in an extended control configuration, theoretical aspects of the generalied plant are addressed in Sect. III. Moreover, based on physical insight, an approach is proposed that exploits the additional actuators and sensors to counteract undesired deformations on the system. In Sect. V, this design approach is formalied in a loop-shaping framework for weighting filter design in H control. Subsequently, this framework is confronted with an industrial high-precision positioning device in Sect. VII, where performance enhancement beyond conventional limitations is successfully shown. Conclusions are drawn in Sect. VIII /$ AACC 984

2 w G w G ext w w ext G ext ext u y C (a) Standard configuration. u y u ext C ext y ext (b) Configuration with additional control inputs and outputs, violating assumptions for synthesis. Fig. 2. Generalied plant configuration. u y C u ext ext y ext (c) Setup corresponding to the proposed design procedure in this paper. II. PROBLEM FORMULATION AND BASIC CONCEPT A. Experimental setup In this paper, the prototype lightweight wafer stage in Fig. is considered. This device is used to position a wafer, on top of which ICs are to be produced, with respect to a light source. To increase productivity, the IC manufacturing industry is currently moving towards the use of wafers with a diameter of 450 mm. Therefore, the wafer stage in Fig. has dimensions mm. Yet, it weighs 3.5 kg only to enable high accelerations. Due to the lightweight design, active control of structural deformations is necessary. B. Problem formulation As is argued in Sect. I, flexible dynamical behavior introduces performance limitations in motion systems. In traditional control configurations, flexible dynamics introduce bandwidth limitations in controlling the rigid-body motion degrees-of-freedom (DOFs) of the wafer stage. The goal of this paper is to go beyond the limits of a traditional control configuration by explicitly accounting for flexible dynamical behavior in control design. In particular, to counteract undesired deformations of the wafer stage, control using a large number of actuators and sensors is investigated. Therefore, the system has been designed to provide abundant opportunities for actuator placement; Lorentactuators can be easily mounted at 8 distinct locations underneath the stage in Fig.. With respect to sensors, linear encoders with nanometer resolution are available for position measurements at all four corners of the stage. Indeed, the use of additional control inputs and outputs provides freedom for performance enhancement, as is formalied next. C. Extra inputs-outputs enable performance improvement In this section, the merits of adding extra inputs and outputs for control are analyed systematically. To this end, the generalied plant configuration in Fig. 2(a) is considered, see, e.g., [3], [8]. Herein, w are exogenous inputs and exogenous outputs of the generalied plant G. The controller C is designed by minimiing a certain norm between w and. Definition (Standard configuration) Consider Fig. 2(a). The performance J achieved by a controller C is given by J (C) := F l (G, C). In addition, the optimal nominal controller C is defined as C = arg min J (C). C Next, the generalied plant is extended with additional inputs u ext and outputs y ext for control, see Fig. 2(b). Definition 2 (Extended configuration) Consider Fig. 2(b), where for G ext it holds that [ ] I 0 0 G 0 I 0 ext I 0 0 I = G. 0 0 The performance J ext that is achieved by a controller C ext in the extended configuration is given by J ext (C ext ) := F l (G ext, C ext ). It is straightforward to show that increasing the number of control inputs and outputs enables performance optimiation. Lemma 3 (Performance enhancement) Consider the generalied plant configurations in Def. and Def. 2. Then, min C ext J ext (C ext ) J (C ). Although conceptually straightforward, the design of a norm-based controller that achieves performance beyond conventional limits is non-trivial. It requires the formulation of a well-posed H synthesis problem, in which the freedom provided by additional inputs and outputs is exploited in judicious weighting filter design. In Sect. III and IV, these aspects are further addressed through the development of a framework for control synthesis, based on Fig. 2(c), in which additional actuators and sensors are exploited. III. THEORETICAL ASPECTS FOR CONTROL SYNTHESIS IN THE EXTENDED GENERALIZED PLANT Although it is clear that additional actuators and sensors enable performance enhancement, cf. Sect. II-C, the configuration in Fig. 2(b) is unsuitable for the synthesis of an optimal controller that exploits new control inputs and outputs in a meaningful way. To further explain this, standard assumptions for H 2 and H optimal control, see [4, Sect. III.A] and [3, Sect ], are reviewed. Assumption 4 Consider the standard generalied plant configuration in Fig. 2(a). Let G have a state-space realiation G = A B B 2 C 0 D 2. C 2 D 2 0 The following assumptions are standard in optimal control. [ ] [ ] Assumption 4.: D2 T C D 2 = 0 I. [ ] [ B 0 Assumption 4.2: D D 2 T =. 2 I] Assumption 4 ensures a bounded control effort in optimal control design. In particular, it follows from Ass. 4. that D 2 985

3 F m F diff x b x 2 x diff Fig. 3. Benchmark system with additional actuator and sensor, which enables a change of the system s stiffness through high-gain control k c. is a tall matrix with full column rank. Hence, the performance outputs = C x + D 2 u include a nonsingular penalty on the control signals u. Similarly, from Ass. 4.2 it follows that D 2 is a wide matrix with full row rank. Thus, the sensors y = C 2 x + D 2 w are all affected by the performance inputs w. Typically, Ass. 4 is violated for the extended generalied plant G ext in Fig. 2(b). In particular, does not include a penalty on the additional control signals u ext, while w does in general not affect the additional sensors y ext. As a consequence, Fig. 2(b) generally leads to an ill-posed optimal control design problem, in which unbounded signals u ext, y ext may result in an attempt to minimie the norm J of the transfer matrix from w to. For well-posed optimal control synthesis, the setup in Fig. 2(c) is considered, where k c k m 2 ext includes a nonsingular penalty on u ext, and w ext affects y ext in a nonsingular way. Under these assumptions, Fig. 2(c) is effectively in the standard format of Fig. 2(a) and enables the exploitation of additional actuators and sensors in optimal control. The setup in Fig. 2(c) extends Fig. 2(b) with auxiliary performance variables w ext ext that in essence serve to bound the control effort of u ext y ext in H synthesis. Herein it is not yet clear how to select w ext and ext, and how to dictate suitable performance goals for these auxiliary channels such that performance beyond conventional limits is achieved. Next, a suitable design philosophy is developed based on physical insight. IV. EXPLOITING PHYSICAL INSIGHT TOWARDS CONTROL DESIGN BEYOND THE CONVENTIONAL LIMITS In this section, physical insight is exploited to develop a design strategy that exploits additional actuators and sensors to go beyond conventional performance limits for flexible motion systems. To develop this insight, the simple model in Fig. 3 is considered. This model is chosen such that it captures all relevant aspects of the wafer stage system in Fig. in view of the development of weighting functions. Here, the position x diff := (x 2 x ) can be measured. In addition, a force F diff := F 2 F can be applied. This leads to the following definition of the extended plant. Definition 5 The extended non-collocated plant is defined as [ ] [ ] [ ] [ ] u y F x2 P ncol,ext : =. u ext y ext F diff x diff A Bode diagram of P ncol,ext is shown in Fig. 4. Herein, 2 P flex : u ext y ext = ms 2 () + 2bs + 2k describes the system s flexible mode. The key point in Fig. 4 P flex Fig. 4. Extended non-collocated plant P ncol,ext (solid) and equivalent plant P ncol,eq (dash-dotted) obtained after closing feedback loop for P flex. F F d + F diff P ncol,ext C flex x diff x 2 P ncol,eq Fig. 5. Equivalent plant P ncol,eq obtained by control of P flex. is that the transfer function F x 2 is non-collated. This leads to performance limitations as robustness has to be accounted for. To anticipate on the design framework in Sect. V, two steps are taken to address conventional performance limitations using the extra plant input and output. i) A controller C flex is designed for the flexible mode P flex. The aim is to enhance the stiffness of the system, such that flexible dynamics prevail at higher frequencies. ii) The equivalent non-collocated plant P ncol,eq is determined, see Fig. 5. This equivalent plant provides new freedom for enhancement of motion performance, e.g., by enabling higher bandwidths using PID-type control. In view of i), the following result is motivated by an investigation of the physics of the benchmark motion system. Result 6 Consider Fig. 5, with static controller C flex = k c. At low frequencies, undesired structural deformations x diff, which result from disturbance forces F d that excite flexible dynamics of the system, are reduced with a factor ( + kc k ). To support Result 6, observe that 2 P flex,cont : F d x diff = ms 2 (2) + 2bs + 2(k + k c ) As a consequence, low-frequent disturbances of magnitude F d result in a deformation x diff F d /(k + k c ), compared to x diff F d /k without the controller C flex, see () and (2). From a physical point of view, an additional stiffness k c is added to the system through control, see Fig. 3. In view of ii), in P ncol,eq, the resonance frequency manifests itself at f res,eq c) 2(k+k m /(2π) H. From a physical point of view, by actively counteracting internal deformations of the system, the controlled system behaves as a rigid-body to a much higher frequency. Indeed, this provides freedom to enhance the control bandwidth. This concept is used in the next section to design weighting filters for H synthesis. V. TRANSLATION TO WEIGHTING FILTERS FOR MULTIVARIABLE H SYNTHESIS In this section, a novel weighting filter design framework for multivariable H control synthesis is proposed, in which 986

4 w + - C s w W P ext W 2 s2 s4 s3 s Ry a3 Rx a a4 a2 Fig. 6. Feedback configuration with P s indicated by dashed box. the philosophy presented in Sect. IV is applied to beat conventional performance limits using additional actuators and sensors. This control design philosophy has a direct interpretation in terms of a desired loop-shape P C. A natural framework to incorporate loop-shape requirements in H synthesis is the shaped-plant framework, [5, Chap. 6]. A. Specifying loop-shape requirements in H control design The shaped-plant framework is based on the feedback configuration in Fig. 6. Herein, P ext is obtained by extending a rigid-body configuration with additional inputs and outputs. Definition 7 The extended plant is defined as [ ] [ ] u y P ext : ũ ỹ, ũ =, ỹ =. u ext y ext Assumption 8 Without loss of generality, inputs and outputs u, y R n are used to control n motion DOFs of the system, whereas u ext, y ext R m are m additional inputs and outputs for control of flexible dynamical behavior, see Sect. IV. Next, P ext is appended with weighting functions W, W 2. Definition 9 The shaped plant is P s = W 2 P ext W. The shaped plant reflects a desired loop-shape that is dictated through W and W 2. Subsequently, a H -controller C s is designed that attains this loop-shape as close as possible while stabiliing the system P s in Fig. 6. This design problem can be cast in the configuration shown in Fig. 2(c). Definition 0 The performance criterion J, used to design the feedback loop in Fig. 6, is given by J (P s, C s ) := T (P s, C s ), where [ ] Ps T (P s, C s ) : w = (I + C I s P s ) [ C s I ]. Using Def. 0, the H -optimal controller C ext for the extended plant P ext is determined by where C ext = W C s W 2, C s = arg min C s J (Ps, C s ), see [5, Chap. 6] for details. In the next section, a framework for the design of weighting filters W and W 2 for highperformance control of flexible motion systems is presented. B. Weighting filter design framework that exploits freedom provided by additional actuators and sensors In this section, the physical insight presented in Sect. IV is integrated into a framework for multivariable control. In essence, the same two steps taken in Sect. IV are followed for weighting filter design, succeeded by a full multivariable H control synthesis. (a) Actuator-sensor configuration. (b) Torsion bending mode. Fig. 7. Inputs and outputs that can be used to compensate for torsion of the wafer stage. Weighting filter design for flexible motion systems D) Consider the flexible behavior P flex : u ext y ext, see Ass. 8. Shape the desired loop-gain of the flexible plant P flex,s = W 2,flex P flex W,flex (3) by designing weighting filters W,flex and W 2,flex. Essentially, W,flex and W 2,flex should incorporate a high gain, hereby dictating a reduction of internal system deformations through active feedback control. This concept is further explained by Result 6 in Sect. II. D2) First, construct the equivalent plant P eq = F l (P ext, W 2,flex W,flex ), (4) which represents the behavior in the motion DOFs under control of the system s flexibilities, see Ass. 8. Subsequently, shape the loop-gain P eq,s = W 2,eq P eq W,eq (5) by designing the weighting filters W,eq, W 2,eq such that new freedom for bandwidth enhancement is exploited. After completing these design steps, the weights W = [ W,eq W,flex], W 2 = [ W2,eq W2,flex] form the shaped extended plant in Def. 9. On the basis of this shaped plant, a H -norm based controller is synthesied along the lines explained in Sect. V-A. In Sect. VII, the proposed design framework is further discussed and illustrated on the industrial lightweight positioning device in Fig.. First, a suitable control configuration for this device is presented in Sect. VI. VI. CONTROL CONFIGURATION AND DYNAMICS OF THE EXPERIMENTAL SETUP In this section, a control configuration for the lightweight wafer stage in Fig. is presented, by selecting suitable actuator and sensor locations. The input-output configuration is chosen in such a way, that a clear illustration of the new concepts presented in this paper is facilitated. A. Actuation principle and actuator-sensor placement State-of-the-art wafer stages are designed to operate in a vacuum. To facilitate this, the system in Fig. is actuated on the basis of magnetic levitation [6]. Hereby, the wafer stage can be positioned contactless. As a result, there are 6 motion degrees-of-freedom (DOFs), i.e., 3 translations [x, y, ] and 3 rotations [R x, R y, R ]. By design, the stroke of the system in the x y plane is limited, as the primary purpose of the prototype is to investigate the effect of structural deformations. Since structural deformations are most manifest in the DOFs [, R x, R y ], see Fig. 7(a), this coordinate frame is considered for control design.. 987

5 ) FRF identification P o Identification data 2) Weighting filter design W, W 2 Validation data 3) Nominal identification ˆP 4) Embedding P o in model set P Fig. 8. Measured frequency response function P o,ext (dotted) and controlrelevant 8th order parametric model ˆPext (solid). Next, an input-output configuration is selected. In this paper, 4 actuators are placed as indicated in Fig. 7(a). Specifically, a, a 2, and a 3 are placed below corners of the stage, whereas a 4 is positioned at the middle of the line between the center and a 2. Moreover, corner sensors s, s 2, and s 3 are used as indicated in Fig. 7(a). Finally, a pieo sensor s 4, which measures strain of the wafer stage, is available at the middle of the line between the center and s 2. Since 4 actuators and sensors are used to control 3 rigid-body DOFs, there is freedom to actively control flexible dynamical behavior of the wafer stage. B. Coordinate frame for extended control design In this section, the input-output behavior of the system is transformed to a coordinate frame that satisfies Ass. 8. Result Let P a,s : a s be the mapping from the system s actuators to sensors, where a = [a, a 2, a 3, a 4 ] T and s = [s, s 2, s 3, s 4 ] T. The extended plant, see Def. 7, is obtained through the decoupling P ext = S y T y P a,s T u S u, where 0 0 T y = , T u = and S u, S y are diagonal scaling matrices , 4 Using the geometry of the wafer stage, see Fig. 7(a), it is immediate that the first three coordinates of P ext contain decoupled rigid-body behavior in the n = 3 motion DOFs [, R x, R y ]. In Fig. 8, collocated behavior is observed in the -translation, whereas non-collocated behavior is observed in the R x - and R y - rotations. After applying rigid-body decoupling, there is m = additional input-output directionality available for control of flexible dynamics. The first flexible mode is a torsion bending of the stage, depicted in Fig. 7(b). This mode can effectively be controlled using the fourth input-output direction of P ext, indicated by P flex in Fig. 8. On the one hand, the fourth output of P ext is the pieo sensor s 4, which measures strain. As a consequence, no rigid-body displacements are observed at this output, while the structural deformations associated with the flexible dynamical behavior of the system P flex 5) Robust control synthesis C RP Fig. 9. Identification and robust control design procedure, where enhanced weighting filter design plays an important role. are measured indeed. On the other hand, the fourth input of P ext does not excite rigid-body behavior, since upon applying a force F = [, 2,, 4] T N to the actuators a, F = F + F 2 + F 3 + F 4 = 0, M Rx = F + F 2 + F F 4 = 0, M Ry = F F 2 + F 3 2 F 4 = 0, i.e., the net force and rotational moments on the stage are ero in the rigid-body coordinates, cf. Fig. 7(a). Before presenting the merits of using the additional control input-output, the essential steps towards synthesis of a robust controller are reviewed. C. Steps towards high-performance robust control design The framework in Fig. 9 describes the essential steps towards synthesis of a high-performance robust controller. First, a frequency response function (FRF) of the true system P o is determined (Step ). On the basis of this FRF, the proposed weighting filter design approach in Sect. V-B can be applied (Step 2). Subsequently, a model set for robust control is constructed, which requires nominal modeling (Step 3) and uncertainty modeling (Step 4). Finally, the optimal robust controller is computed (Step 5). The system identification for robust control design procedure in [2] is employed. Herein, the performance criterion J in Def. 0 is jointly optimied during the modeling step and the robust control synthesis step. In particular, robust-controlrelevant identification is pursued, see [2] for further details. An important implication of this approach is that the novel weighting filter design framework influences all subsequent design steps towards the synthesis of a robust controller. VII. BEYOND CONVENTIONAL BANDWIDTH LIMITATIONS USING THE EXTRA ACTUATOR-SENSOR PAIR In this section, the framework for weighting filter design in Sect. V-B is confronted with the prototype wafer stage. The aim is to beat conventional bandwidth limits. Step. Frequency response function identification The identified frequency response function P o,ext of the system in Fig. is depicted in Fig. 8. The torsion bending mode of the stage manifests itself as a lightly damped resonance phenomenon at 43 H, whereas remaining flexible phenomena occur above 500 H. Due to the actuator-sensor configuration in Fig. 7(a), noncollocated behavior is observed beyond the torsion frequency in the rotational coordinates R x and R y. Hence, the torsion mode imposes restrictions on the achievable control bandwidths using conventional rigid-body control. 988

6 Fig. 0. Phase (deg) Magnitude of W 2,flex W,flex = k (dotted) and C flex (solid) Fig.. Bode diagram of the loop-gain P flex C flex. Step 2. Application of the new design framework In this section, weighting filters are designed for robust control, following the two steps in Sect. V. First, the stiffness of the wafer stage in the direction of the torsion bending mode is enlarged. Second, the equivalent plant in the motion DOFs [, R x, R y ] is determined, for which the obtained freedom for bandwidth enhancement is exploited. Design step D In design step D, Sect. V-B, the additional control input and output available on top of the rigid-body configuration are exploited to control the flexible dynamical behavior P flex. As observed in Fig. 8, below 200 H, P flex m s 2 + d s + k where the compliance /k is approximately -45 db, i.e., k N/m. By controlling P flex with a high loopgain, the apparent stiffness of the system can be enhanced, which reduces internal deformations, cf. Result 6 in Sect. II. Here, it is aimed to double the stiffness of the torsion mode. Thus, for P flex,s in (3), the weights W,flex = W 2,flex = k = , (6) dictate the desired loop-gain for the torsion mode. For the considered control configuration, it is challenging to achieve the specified gain in the torsion loop. If P flex is multiplied with a gain k of 45 db, then the torsion mode at 43 H, as well as all parasitic higher order flexible modes beyond 500 H, will have a loop-gain that is lifted above the 0 db line, see Fig. 8. Hence, all these phenomena become relevant for stability. To achieve a stable high-gain torsion loop using robust control synthesis, a non-conservative model set is required that encompasses all flexible phenomena with little uncertainty. This demands for an accurate nominal model that captures all these phenomena. Due to the involved modeling complexity, this is beyond the scope of this paper. Instead, the torsion feedback loop is designed on the basis of manual loop-shaping. The resulting controller is depicted in Fig. 0. Even using SISO manual design, it is non-trivial to generate Fig. 2. Equivalent plant P o,eq (dotted) and parametric fit ˆPeq (solid) under active control of the torsion loop. The initial model of the motion DOFs [, R x, R y] from Fig. 8 is shown for comparison (dashed). sufficient phase-lead around 200 H while at the same time achieving sufficient suppression of higher order modes using notches and roll-off, which is needed to enable a stable highgain torsion loop. Design step D2 In design step D2, Sect. V-B, the FRF of the equivalent plant under control of the torsion loop is determined. Instead of (4), P eq = F l (P o,ext, C flex ) is determined by means of a system identification experiment, where the manually designed controller C flex is implemented on the true system. The FRF is shown in Fig. 2. Result 2 By active control of the torsion mode, the mechanical stiffness of the stage is successfully modified. As a result, the frequency at which the torsion mode manifests itself shifts from 43 to 93 H. Theoretically, under proportional control of the torsion mode with a gain k = , see (6), the resonance frequency would manifest itself at 2 43 = 202 H. The manually designed torsion controller C flex achieves a shift of the resonance frequency that is slightly lower only. Also, the controller adds damping to the torsion mode, as is clearly observed in Fig. 2. Now that new freedom has been generated for bandwidth enhancement in the motion DOFs [, R x, R y ], this freedom should be exploited in the design of weighting filters W,eq and W 2,eq, see (5). These weighting filters are designed along the lines of [2, Sect. III.A] and [5], and reflect common loop-shaping goals for PID-type of motion control as also encountered in, e.g., [], [7], [0]. To obtain a meaningful robust performance optimiation problem, it is needed to dictate attainable control goals. On the other hand, it is desired to achieve control bandwidths that are as high as possible. By using a procedure similar to [8], target bandwidths of [64, 55, 56] H for [, R x, R y ] are selected. Remark on notation for 3 3 robust control design As motivated in design step D, instead of performing a formal 4 4 robust control design for P ext in Fig. 8, the controller for P flex is designed using manual loop-shaping. It then remains to design a 3 3 robust controller for P eq in Fig

7 Fig. 3. Comparison of the robust controllers in motion DOFs [, R x, R y] for conventional plant P o (dashed) and equivalent plant P eq,o (solid). Amplitude (m).5 x Time (s) Fig. 4. Standstill error on application of a disturbance signal: conventional (blue) and extended (green) control configuration. Step 3-5. Modeling and robust control design Next, step 3-5 in Fig. 9 are performed using the framework in [2]. The robust controller is synthesied for the equivalent plant under control of the torsion mode, with target bandwidths of [64, 55, 56] H, see Step 2. D-K iterations is used to compute the optimal robust controller, see Fig. 3. To evaluate the merits of the proposed design procedure, a comparison is made with conventional rigid-body control. Therefore, a 3 3 robust controller is designed for the first 3 input-output directions of P o,ext in Fig. 8. It is emphasied that no control of the torsion mode is applied. The same steps as outlined in Fig. 9 are followed. In the weighting filter design, again a procedure similar to [8], [9] is used to establish meaningful target bandwidths for [, R x, R y ] of [54, 44, 44] H. The rigid-body controller is shown in Fig. 3. Result 3 The conventional robust rigid-body controller achieves bandwidths of [50, 30, 30] H, which is in line with the results in [20] on the same setup. By enhancing the stiffness of the wafer stage through active control of the torsion mode, a bandwidth enhancement towards [60, 39, 39] H is successfully obtained. Time domain results with the controllers implemented, see Fig. 4, corroborate the observations, i.e., the extended control configuration significantly enhances the performance. VIII. CONCLUSIONS Control performance in traditional motion systems is limited by flexible dynamical behavior. In this paper, a new framework is proposed that enables high performance robust motion control design. The framework exploits additional inputs and outputs. A systematic weighting function design procedure for H control design is presented and embedded in an identification and robust control design framework. The approach is demonstrated on an industrial wafer stage system, confirming a significant performance enhancement. ACKNOWLEDGEMENTS The authors gratefully acknowledge the contributions by Okko Bosgra in an early stage of this research. This research is supported by ASML research and by the Innovational Research Incentives Scheme under the VENI grant Precision Motion: Beyond the Nanometer (no. 3073) awarded by NWO (The Netherlands Organisation for Scientific Research) and STW (Dutch Science Foundation). REFERENCES [] V. M. Martine and T. F. Edgar, Control of lithography in semiconductor manufacturing, IEEE Control Systems Magaine, vol. 26, no. 6, pp , [2] T. Oomen, R. van Herpen, S. Quist, M. van de Wal, O. Bosgra, and M. 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