Control Method and Control System for Controlling the Position of an Object Using an Electromagnetic Actuator

Abstract

The present invention provides a control method for controlling the position of an object using an electromagnetic actuator. The method includes determining a position control error between a desired position and an actual position of the object, determining a feedback control signal based on the position control error, determining a feedforward control signal based on the desired position, combining the feedback control signal and the feedforward control signal into an actuator input, determining an actuator gain correction based on the actuator input and the actual position of the object, applying the actuator gain correction to the actuator input to provide a corrected actuator input, and sending the corrected actuator input to the electromagnetic actuator to exert an actuator force on the object.

Description

Technical Field

[0001]

[0001] The present invention relates to a control method for controlling the position of an object using an electromagnetic actuator, and a control system for controlling the position of an object. The present invention further relates to a lithographic apparatus comprising such a control system.

Background Art

[0002]

[0002] A lithographic apparatus is a machine constructed to apply a desired pattern onto a substrate. The lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). The lithographic apparatus can project a pattern onto a layer of radiation-sensitive material (resist) provided on a substrate, for example, in a patterning device (e.g., a mask).

[0003]

[0003] To project a pattern onto a substrate, the lithographic apparatus can use electromagnetic radiation. The wavelength of this radiation determines the minimum size of the features that can be formed on the substrate. A lithographic apparatus using extreme ultraviolet (EUV) radiation having a wavelength in the range of 4 - 20 nm, for example 6.7 nm or 13.5 nm, can be used to form smaller features on a substrate than, for example, a lithographic apparatus using radiation with a wavelength of 193 nm.

[0004]

[0004] In known embodiments of a lithographic apparatus, a projection system comprising one or more mirrors can be provided. The mirrors can be position-controlled, for example, to accurately position the mirrors so as to cancel vibrations. Such position-controlled mirrors can be actively controlled in a control system comprising a feed-forward controller and a feedback controller. An electromagnetic actuator can be used to exert an actuating force on the mirror based on control signals provided by the feed-forward controller and the feedback controller.

[0005]

[0005] The operation of an electromagnetic actuator, e.g., the force provided by the electromagnetic actuator in response to a current sent into the electromagnetic actuator, may depend on a reference, e.g., the actual actuator position with respect to a reference frame. Thus, the characteristics of each electromagnetic actuator are calibrated before use. During calibration of the electromagnetic actuator, the non-linearity in the operation of the electromagnetic actuator can be determined for various positions of the electromagnetic actuator. Based on the determined operation, a non-linear compensation according to the input current and the actual actuator position can be applied in the control system to provide an adjusted current signal to the electromagnetic actuator.

[0006]

[0006] However, due to slow changes over time of variables such as reference drift, and slow aging deterioration of the magnet, the non-linearity correction obtained by calibration is no longer accurate after a certain period, which can lead to changes in the offset current. In particular, when non-linearity calibration cannot be performed within the machine, any unknown changes over time cannot be compensated. In fact, this can be partially solved by in-machine local actuator gain calibration at multiple positions, but it takes a significantly large amount of time.

[0007]

[0007] Further, the calibration accuracy of the electromagnetic actuator is limited. This accuracy may be insufficient to meet the current accuracy specifications. Further in the future, higher accelerations of the mirror may be required, and the accuracy specifications for the electromagnetic actuator can also be improved.

Summary of the Invention

[0008]

[0008] The object of the present invention is to provide a control method for controlling the position of an object using an electromagnetic actuator, which can provide enhanced accuracy in object positioning. In particular, the object of the present invention is to provide a control method for controlling the position of an object using an electromagnetic actuator, which takes into account the drift in the operation of the electromagnetic actuator. Further, the object of the present invention is to provide a control system for controlling the position of an object, such as a mirror of a projection system of a lithographic apparatus, the control system being arranged to provide improved accuracy in object positioning. In particular, the control system may take into account the drift in the operation of the electromagnetic actuator.

[0009]

[0009] According to an aspect of the present invention, there is provided a control method for controlling the position of an object using an electromagnetic actuator, the method comprising: determining a position control error between a desired position and an actual position of the object; determining a feedback control signal based on the position control error; determining a feedforward control signal based on the desired position; combining the feedback control signal and the feedforward control signal to an actuator input; determining an actuator gain correction based on the actuator input and the actual position of the object; applying the actuator gain correction to the actuator input to provide a corrected actuator input; and sending the corrected actuator input to the electromagnetic actuator to exert an actuator force on the object. comprises.

[0010]

[0010] According to an aspect of the present invention, there is provided a control system for controlling the position of an object, the control system comprising: an electromagnetic actuator arranged to exert an actuator force on the object based on an actuator input; A feedback control device arranged to provide a feedback control signal based on a position control error between a desired position and an actual position of an object, and A feedforward control device arranged to provide a feedforward control signal based on the desired position, and Comprising A control system arranged to combine the feedback control signal and the feedforward control signal into an actuator input, The control system comprises an actuator gain correction device, which is arranged to determine an actuator gain correction based on the actuator input and the actual position of the object, and to apply the actuator gain correction to the actuator input to provide a corrected actuator input to the electromagnetic actuator.

[0011]

[0011] According to an aspect of the invention, there is provided a lithographic apparatus comprising such a control system for controlling the position of an object of the lithographic apparatus.

[0012]

[0012] Next, embodiments of the invention will be described by way of example only with reference to the accompanying schematic diagrams.

Brief Description of the Drawings

[0013]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

[0014]

[0013] FIG. 1 shows a lithography system including a radiation source SO and a lithography apparatus LA. The radiation source SO is configured to generate an EUV radiation beam B and supply the EUV radiation beam B to the lithography apparatus LA. The lithography apparatus LA includes an illumination system IL, a support structure MT configured to support a patterning device MA (e.g., a mask), a projection system PS, and a substrate table WT configured to support a substrate W.

[0015]

[0014] A substrate table positioning system WTP is provided to position the substrate table WT at a desired position. The substrate positioning system WTP includes a position measurement system for measuring the position of the substrate table WT and an actuation system for moving the substrate table WT to a desired position. A patterning device support positioning system MTP is provided to position the support structure MT at a desired position. The patterning device support positioning system MTP also includes a position measurement system for measuring the position of the support structure MT and an actuation system for moving the support structure MT to a desired position.

[0016]

[0015] The illumination system IL is configured to condition the EUV radiation beam B before the EUV radiation beam B is incident on the patterning device MA. In addition to this, the illumination system IL may include a facet field mirror device 10 and a facet pupil mirror device 11. The facet field mirror device 10 and the facet pupil mirror device 11 provide the EUV radiation beam B with a desired cross-sectional shape and a desired intensity distribution. The illumination system IL may include other mirrors or devices in addition to or instead of the facet field mirror device 10 and the facet pupil mirror device 11.

[0017]

[0016] After being conditioned in this way, the EUV radiation beam B interacts with the patterning device MA. As a result of this interaction, a patterned EUV radiation beam B’ is generated. The projection system PS is configured to project the patterned EUV radiation beam B’ onto the substrate W. For this purpose, the projection system PS may comprise a plurality of mirrors 13, 14 configured to project the patterned EUV radiation beam B’ onto the substrate W held by the substrate table WT. The projection system PS may apply a reduction factor to the patterned EUV radiation beam B’, and thus form an image with features smaller than the corresponding features on the patterning device MA. For example, a reduction factor of 4 or 8 may be applied. The projection system PS is shown in FIG. 1 as having only two mirrors 13, 14, but may include a different number of mirrors (for example, six or eight mirrors).

[0018]

[0017] The substrate W may include a previously formed pattern. In this case, the lithographic apparatus LA aligns the image formed by the patterned EUV radiation beam B’ with the previously formed pattern on the substrate W.

[0019]

[0018] A relative vacuum, i.e. a small amount of gas (e.g. hydrogen) at a pressure much lower than atmospheric pressure, may be provided within the radiation source SO, within the illumination system IL and / or within the projection system PS.

[0020]

[0019] The radiation source SO can be a laser-produced plasma (LPP) source, a discharge-produced plasma (DPP) source, a free electron laser (FEL), or any other radiation source capable of generating EUV radiation.

[0021]

[0020] The lithography process includes a series of projection phases in which the patterned EUV radiation beam B’ is projected onto the substrate W (exposure phase) and / or the substrate W is aligned with the patterned EUV radiation beam B’ (alignment phase), and an idle phase in which the patterned EUV radiation beam B’ is not projected onto the substrate W or an irrelevant part of the substrate W and the positioning accuracy of the substrate W with respect to the patterned EUV radiation beam B’ is not so critical. During the projection phases, the patterning device and the substrate can be moved in a scanning motion at a constant scan speed. The idle phase can be used to decelerate and (re)accelerate the patterning device MT and the substrate W to the desired scan speed and the desired alignment with respect to the EUV radiation beam B and the patterned EUV radiation beam B’, respectively. The constant scan speed of the patterning device MT is usually different from the constant scan speed of the substrate W.

[0022]

[0021] FIG. 2 shows the projection system PS of the lithographic apparatus of FIG. 1 in more detail. The projection system PS comprises mirrors 13, 14, hereinafter referred to as the first mirror 13 and the second mirror 14. The first mirror 13 and the second mirror 14 are position-controlled mirrors. This means that the position of the first mirror 13 is controlled within a first range of motion and the position of the second mirror 14 is controlled within a second range of motion.

[0023]

[0022] The first position measurement system 15 is provided for measuring the actual position of the first mirror 13 relative to a reference frame 17, for example a sensor frame. The position measurement system 15 is, for example, an interferometer position measurement system. The control unit CON is provided for controlling the position of the first mirror 13 relative to the reference frame 17. Based on the output signal of the control unit CON, an electromagnetic actuator 16, for example a Lorentz actuator, is used to position the first mirror 13 within a desired position relative to the reference frame 17. In the illustrated embodiment, the electromagnetic actuator 16 is mounted on a force frame 18. The position of the force frame 18 can vary, for example, by drifting relative to the position of the reference frame 17.

[0024]

[0023] The control unit CON, the position measurement system 15, and the electromagnetic actuator 16 form a control system arranged to control the position of the first mirror 13 in a plurality of degrees of freedom, for example six degrees of freedom.

[0025]

[0024] Similarly, in order to control the position of the second mirror 14 of the projection system PS and other position control mirrors, further position measurement systems for measuring the position of the mirror can be provided for the second mirror 14 and each of the other mirrors, respectively. The control unit CON can comprise a control loop for controlling the position of the second mirror 14 and the other mirrors. The control unit CON can provide an output signal to each actuator in order to position the second and other mirrors in a desired position.

[0026]

[0025] The control unit CON is shown as a single unit for controlling the position of the first mirror 13, the second mirror 14, and the other mirrors. This control unit CON can be integrated, for example, into a central processing device of the lithographic apparatus LA or a separate control unit of the projection system PS.

[0027]

[0026] FIG. 3 shows a control method of a control system that controls the position of the first mirror 13 in six degrees of freedom using the electromagnetic actuator 16. Block P represents the electromagnetic actuator 16, the first mirror 13, and the position measurement system 15 as shown in FIG. 2. The input of block P is the actuator input sent to the electromagnetic actuator 16, and the output of block P is the actual position x of the first mirror 13 measured by the position measurement system 15.

[0028]

[0027] The actual position x of the first mirror 13 can be compared with the desired position r of the first mirror 13 in a comparison device to obtain a position control error e. The desired position can be provided by a setpoint generator. The position control error e is sent to a feedback control device FBC, such as a PID controller, to obtain a feedback control signal f fb In addition, the desired position r is sent to a feedforward control device FFC. Based on the desired position r, the feedforward control device FFC provides a feedforward control signal f ff

[0029]

[0028] The feedback control signal f fb and the feedforward control signal f ff are combined into a control signal and sent to a decoupling device T f to convert the control signal f in the control coordinates into an actuator input a in the actuator coordinates. The decoupling device T f is a two-stage decoupling system including gain balancing and gain scheduling steps.

[0030]

[0029] The gain balancing and gain scheduling steps provide a conversion of a control signal, such as a current setpoint for the electromagnetic actuator 16, using, for example, a non-linear function of motor constant compensation, resulting in a desired actuator force.

[0031]

[0030] In one embodiment, the gain balancing and gain scheduling steps may, for example, apply a series of linear controllers, each linear controller being arranged to provide specific control for different operating points of the system, such as the position of the electromagnetic actuator 16. Based on one or more scheduling variables, the actual operating range of the electromagnetic actuator can be determined, an associated linear controller can be selected, and respective control signals are provided. In an alternative embodiment, the multi-position gain balancing can be extended using specific filtering to address position dependence.

[0032]

[0031] The actuator input a is sent to the electromagnetic actuator 16 to move the first mirror 13 towards the desired position.

[0033]

[0032] The control system shown in FIG. 3 may be suitable for controlling the position of the well-calibrated electromagnetic actuator 16. The calibration data for the calibration of the electromagnetic actuator 16 may be included in the conversion of the control signal f into the actuator input, as applied by the decoupling device T f .

[0034]

[0033] However, due to slow changes over time of variables such as reference drift and slow aging of the magnets, the non-linearity correction obtained by calibration is no longer accurate after a certain period and can lead to changes in the offset current. In particular, when non-linearity calibration cannot be performed within the machine, any unknown changes over time cannot be compensated. In practice, this can be partially solved by local actuator gain calibration within the machine at multiple positions, but this requires a significant amount of time, which is usually not desirable.

[0035]

[0034] Figure 4 shows an embodiment of a control method of a control system provided with an actuator gain correction device arranged to take into account these gradual changes over time of the variables by applying an actuator gain correction K to an actuator signal a. The actuator gain correction K applied to the actuator input a is shown as block K in Figure 4, and the corrected actuator input a is sent to each electromagnetic actuator 16 c is provided.

[0036]

[0035] In the actuator gain correction device, the actuator gain correction K is calculated based on a feedforward control signal f ff and a position control error e.

[0037]

[0036] The actuator gain correction device includes a decoupling device T fgc , a conversion matrix device T e , a gain correction calculation device θ, and a gain correction block K provided to apply the actuator gain correction K to the actuator input a to obtain the corrected actuator input a c .

[0038]

[0037] The feedforward control signal f ff is sent to the decoupling device T ff to convert the feedforward control signal f ffa in the control coordinates into the feedforward control signal f fgc in the actuator coordinates. The decoupling device T fgc applies the same conversion as the decoupling device T f of the main control loop, that is, the decoupling device T fgc is a two-stage decoupling system including the same decoupling steps as the decoupling device T f .

[0039]

[0038] Therefore, the feedforward control signal f ffacan be determined as follows.

[0039] f ffa =T fgc .f ff In the above equation, T fgc is the force conversion matrix from the feedforward force in the control coordinates to the actuator coordinates, and f ff is the feedforward control signal.

[0040]

[0040] The position control error e is the conversion matrix device T a to convert the position control error e in the control coordinates to the position control error e in the actuator coordinates e is sent to.

[0041]

[0041] The position control error in the actuator coordinates can be determined as follows. e a =T e .e In the above equation, T e is the position conversion matrix from the object position coordinates to the actuator coordinates, and e is the position control error.

[0042]

[0042] In the gain correction calculation device θ, the value of the actuator gain correction K is based on the position control error e a and the feedforward control signal f ffa in the actuator coordinates. The actuator gain correction K can be calculated, for example, as follows. K=-Γ∫(p 11 e a (t)+p 12 (de a (t)) / dt)f ffa (t)dt In the above equation, K is the actuator gain correction, p 11 is the first constant, e a is the position control error in the actuator coordinates, p 12 is the second constant, and f ffa is the feedforward control signal in the actuator coordinates.

[0043]

[0043] Figure 5 shows the calculation of the actuator gain correction K within the gain correction calculation device θ in block form. The constants can have any suitable value that ensures the stability of the actuator gain correction device. Advantageously, the first constant p11 can be selected to be substantially equal to the proportional control constant of the feedback control device FBC, and the second constant p12 can be selected to be substantially equal to the derivative control constant of the feedback control device FBC. The term p 11 e a (t) + p 12 (de a (t)) / dt) can be, or at least partially, already calculated within the feedback control device FBC and can be obtained directly from the feedback control device FBC.

[0044]

[0044] The calculated gain correction K is applied in the main control block K to adjust the actuator input a to the corrected actuator input a c to take into account the gradual change over time of variables that affect the operation of the electromagnetic actuator, such as reference drift, and the gradual aging of the gravity compensator magnet. c The corrected actuator input a

[0045]

[0045] Since the gain correction K is calculated in actuator coordinates, in a position control system configured to control position in six degrees of freedom, only six compensation parameters need to be calculated. In this way, appropriate compensation can be efficiently obtained with efficient calculation.

[0046]

[0046] In general, an adaptive scheme such as that shown in Figure 5 may be sensitive to noise due to the multiplication x of the position control error e a and the feedforward control signal f ffa which may have a coherent noise contribution. However, this problem is avoided in this adaptive scheme by using a feedforward control signal f ffa without noise.

[0047]

[0047] Further, in the actuator gain correction device shown in FIGS. 4 and 5, the position control error e within the actuator coordinates a and the feedforward control signal f within the actuator coordinates ffa are used to calculate the actuator gain correction K. Additionally or alternatively, parameters based on the position control error e a such as filtered values, derivative values, integral values, and / or combinations thereof, and the feedforward control signal f ffa can be used to calculate the actuator gain correction K.

[0048]

[0048] FIG. 6 shows an alternative control method having an actuator gain correction device for applying the gain correction K to the actuator input a of the electromagnetic actuator of the control system. The main control method corresponds to the control method of FIG. 3. In the actuator gain correction device of FIG. 6, the gain correction K is determined based on the feedforward control signal f ff and the feedback control signal f ff .

[0049]

[0049] The feedforward control signal f ff is sent to the decoupling device T ffa to convert the feedforward control signal within the control coordinates to the feedforward control signal f fgc within the actuator coordinates. T f and T fgc apply the same decoupling step. The feedback control signal f fb is sent to the conversion matrix device T fb to convert the feedback control signal within the control coordinates to the feedback control signal f fba within the actuator coordinates. c

[0050]

[0050] The feedforward control signal f within the actuator coordinates ffa ​and the feedback control signal f within the actuator coordinates fba is sent to the gain correction calculation device θ2.

[0051]

[0051] Figure 7 shows the block method of the gain correction calculation device θ2. The calculation of the gain correction K in this method is as follows. K = -Γf fba f ffa (t)dt In the above formula, K is the actuator gain correction, and f fba is the feedforward control signal within the actuator coordinates, and f ffa is the feedforward control signal within the actuator coordinates.

[0052]

[0052] The adaptive method of the actuator gain correction device in FIGS. 6 and 7 is based on the output of the feedback control device FBC, that is, the feedback control signal f fb This value is directly obtained from the feedback control device FBC. However, the presence of the integral operation of the PID controller of the feedback control device FBC within the feedback control signal f fb should be considered in this alternative embodiment.

[0053]

[0053] In addition to or as an alternative to the feedforward control signal f ffa and the feedback control signal f fba parameters based on these values such as filtered values, derivative values, integral values, and / or combinations thereof can be used to calculate the actuator gain correction K.

[0054]

[0054] The actuator gain correction device shown in FIGS. 4 to 7 can take into account the gradual change over time of the variables related to the operation of the electromagnetic actuator without the need for recalibration for non-linear correction. This eliminates the need for the significantly large amount of time usually required for recalibration.

[0055]

[0055] A control method with an actuator gain correction device enables a higher mirror acceleration, for example, when the source power increases.

[0056]

[0056] FIG. 8 shows another embodiment of a control method with an actuator gain correction device. The main control method substantially coincides with the control method of FIG. 3 by adding an actuator gain correction K. This actuator gain correction K is applied to the actuator input a to obtain a corrected actuator input a. c

[0057]

[0057] In the actuator gain correction device of this control method, the gain correction K is determined based on the actuator input a and the actual position x of the first mirror 13.

[0058]

[0058] The actuator gain correction device is arranged to reconstruct the actual corrected actuator input a based on the actual position x of the first mirror 13. cr a cr =Ts 2 x In the above formula, T is the conversion from the actual position to the actuator input, and x is the actual position of the first mirror 13.

[0059]

[0059] This reconstructed actual corrected input a cr is compared with an estimated corrected actuator input a that can be obtained by applying an estimated actuator gain correction K to the actuator a. The comparison between a and a results in a difference between the reconstructed actual corrected input a and the estimated corrected actuator input a. This difference is the estimation error g. e e a cr and a e cr e

[0060]

[0060] The estimated actuator gain correction K e ​​​​​​The recursive least squares method can be used to calculate the estimation, and this estimated actuator gain correction K e can be used as the actuator gain correction K in the actual control loop. The estimated actuator gain correction K e can always be updated based on the estimation error ε at time t and the estimated actuator gain correction K at time t - 1 e .

[0061]

[0061] In the above, various embodiments have been described in which the actuator gain correction is determined based on at least one control value. In the embodiment of FIG. 4, the actuator gain correction is based on the position control error e and the feedforward control signal f ff . In the embodiment of FIG. 6, the actuator gain correction is based on the feedback control signal f fb and the feedforward control signal f ff . In the embodiment of FIG. 8, the actuator gain correction is determined based on the actuator input a and the actual position x. In other embodiments, other control values, i.e., other values currently available within the control system, can be used to determine the actuator gain correction that is applied to the actuator input a to obtain the corrected actuator input a c . These control values include, for example, the desired position r, the position control error e, the feedforward control signal f ff , the feedback control signal f fb , the control signal f, the actuator input a, the corrected actuator input a c , and the actual position x.

[0062] [

[0062] ] Above, a position control system for controlling the position of the mirrors of a projection system in six degrees of freedom has been described. The position control system can also be used, for example, in a control system having six or more electromagnetic actuators for positioning an object in six degrees of freedom, to control the position of another object that is positioned with at least one electromagnetic actuator in a plurality of degrees of freedom. Such objects are, for example, a substrate support or a patterning device support.

[0063] [

[0063] ] Although this document specifically refers to the use of a lithographic apparatus in the manufacture of ICs, it should be understood that the lithographic apparatus described herein may have other applications. Possible other applications include the manufacture of integrated optical systems, guidance and detection patterns for magnetic domain memories, flat panel displays, liquid crystal displays (LCDs), thin film magnetic heads, and the like.

[0064] [

[0064] ] Although this document specifically refers to the use of embodiments of the present invention in the context of a lithographic apparatus, embodiments of the present invention may be used in other apparatuses. Embodiments of the present invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus for measuring or processing an object such as a wafer (or other substrate) or a mask (or other patterning device). These apparatuses may generally be referred to as lithographic tools. Such lithographic tools may use vacuum conditions or atmospheric (non-vacuum) conditions.

[0065] [

[0065] ] Above, it has been specifically mentioned that embodiments of the present invention are used in the context of optical lithography. As long as the context permits, it should be understood that the present invention is not limited to optical lithography and may also be used in other applications such as imprint lithography.

[0066]

[0066] Although specific embodiments of the present invention have been described above, it will be understood that the present invention can be implemented in ways other than those described. The above description is intended to be illustrative and not limiting. Thus, it will be apparent to those skilled in the art that changes can be made to the present invention as described without departing from the scope of the claims set forth below. Other aspects of the present invention are set forth in the numbered clauses below.

[0067] 1. A control method for controlling the position of an object using an electromagnetic actuator, comprising: determining a position control error between a desired position and an actual position of the object; determining a feedback control signal based on the position control error; determining a feedforward control signal based on the desired position; combining the feedback control signal and the feedforward control signal into an actuator input; determining an actuator gain correction based on the actuator input and the actual position of the object; applying the actuator gain correction to the actuator input to provide a corrected actuator input; and sending the corrected actuator input to the electromagnetic actuator to exert an actuator force on the object. The control method as described above. 2. The control method according to clause 1, wherein combining the feedback control signal and the feedforward control signal into the actuator input includes a decoupling step. 3. The control method according to clause 1 or 2, wherein determining the actuator gain correction is based on the feedforward control signal and the position control error or the feedback control signal. 4. The actuator gain correction is determined as follows: K=-Γ∫(p 11 e a (t)+p 12 (de a (t)) / dt)fffa (t)dt In the above equation, K is the actuator gain correction, and p 11 is the first constant, and e a is the position control error within the actuator coordinates, and p 12 is the second constant, and f ffa is the feed-forward control signal within the actuator coordinates. The control method according to any one of clauses 1 to 3. 5. The position control error within the actuator coordinates is determined as follows: e a =T e .e In the above equation, T e is the position transformation matrix from the object position coordinates to the actuator coordinates, and e is the position control error. The control method according to clause 4. 6. The feed-forward control signal within the actuator coordinates is determined as follows: f ffa =T fgc .f ff In the above equation, T fgc is the transformation matrix from the feed-forward force within the control coordinates to the actuator coordinates, and f ff is the feed-forward control signal. The control method according to clause 4 or 5. 7. Combining the feedback control signal and the feed-forward control signal into the actuator input includes a decoupling step, and T fgc corresponds to the decoupling step. The control method according to clause 6. 8. Determining the feedback control signal is based on the proportional control constant and the derivative control constant. p 11 is substantially equal to the proportional control constant, and p 12 is substantially equal to the derivative control constant. The control method according to any one of clauses 4 to 7. 9. The method is reconstructing the actual corrected actuator input based on the actual position of the object, and Estimating actuator gain correction based on the difference between the actuator input and the corrected actuator input. The control method according to any one of clauses 1 or 2, including this. 10. The control method according to any one of clauses 1 to 9, wherein the method is arranged to control the position of the object in six degrees of freedom. 11. The control method according to any one of clauses 1 to 10, wherein the object is an optical element of the projection system of a lithographic apparatus. 12. A control system for controlling the position of an object, the control system comprising: An electromagnetic actuator arranged to exert an actuator force on the object based on an actuator input; A feedback control device arranged to provide a feedback control signal based on a position control error between a desired position and an actual position of the object; A feedforward control device arranged to provide a feedforward control signal based on the desired position; Comprising; The control system is arranged to combine the feedback control signal and the feedforward control signal into the actuator input. The control system comprises an actuator gain correction device, the actuator gain correction device being arranged to determine an actuator gain correction based on the actuator input and the actual position of the object, and to apply the actuator gain correction to the actuator input to provide a corrected actuator input to the electromagnetic actuator. Control system. 13. The control system according to clause 12, wherein the control system comprises a decoupling device for applying a decoupling step to the combined feedback control signal and feedforward control signal. 14. The control system according to clause 12 or 13, based on the feedforward control signal and the position control error or the feedback control signal. 15. The actuator gain correction device is arranged to determine the actuator gain correction as follows: K=-Γ∫(p 11 e a (t)+p 12 (de a (t)) / dt)f ffa (t)dt In the above equation, K is the actuator gain correction, p 11 is the first constant, e a is the position control error within the actuator coordinates, p 12 is the second constant, f ffa is the feedforward control signal within the actuator coordinates. The control system according to any one of clauses 12 to 14. 16. The actuator gain correction device is arranged to determine the position error within the actuator coordinates as follows: e a =T e .e In the above equation, T e is the position conversion matrix from the object position coordinates to the actuator coordinates, and e is the position control error. The control system according to clause 15. 17. The actuator gain correction device is arranged to determine the feedforward control signal within the actuator coordinates as follows: f ffa =T fgc .f ff In the above equation, T fgc is the force conversion matrix from the feedforward force within the control coordinates to the actuator coordinates, and F ff is the feedforward control signal. The control system according to clause 15 or 16. 18. Combining the feedback control signal and the feedforward control signal into the actuator input includes a decoupling step, and T fgc corresponds to the decoupling step. The control system according to clause 17. 19. The feedback control device includes a proportional control constant and a derivative control constant, where p 11 is substantially equal to the proportional control constant, and p 12 is substantially equal to the derivative control constant, and the control system according to any one of clauses 15 to 18. 20. The actuator gain correction device is configured to reconstruct the actually corrected actuator input based on the actual position of the object, and configured to estimate the actuator gain correction based on the difference between the actuator input and the corrected actuator input, and is arranged in the control system according to clause 12 or 13. 21. The control system is arranged to control the position of the object with six degrees of freedom, and the control system according to any one of clauses 12 to 20. 22. The control system includes a position measurement system for determining the actual position of the object, and the control system according to any one of clauses 12 to 21. 23. A lithographic apparatus comprising a control system according to any one of clauses 12 to 22 for controlling the position of an object of the lithographic apparatus. 24. The object is an optical element of the lithographic apparatus, and the lithographic apparatus according to clause 23. 25. The object is a mirror of the projection system of the lithographic apparatus, and the lithographic apparatus according to clause 23 or 24.

Claims

1. A control method for controlling the position of an object using an electromagnetic actuator, comprising: determining a position control error between a desired position and an actual position of the object; determining a feedback control signal based on the position control error; determining a feedforward control signal based on the desired position; combining the feedback control signal and the feedforward control signal into an actuator input; determining an actuator gain correction based on the actuator input and the actual position of the object; applying the actuator gain correction to the actuator input to provide a corrected actuator input; and sending the corrected actuator input to the electromagnetic actuator to exert an actuator force on the object. A control method as described above.

2. The control method according to claim 1, wherein combining the feedback control signal and the feedforward control signal into the actuator input includes a decoupling step.

3. The control method according to claim 1 or 2, wherein determining the actuator gain correction is based on the feedforward control signal and the position control error or the feedback control signal.

4. The actuator gain correction is determined as follows: K = -Γ∫(p １１ e ａ (t) + p １２ (de ａ (t)) / dt)f ｆｆａ (t)dt In the above formula, K is the actuator gain correction, and p １１ is the first constant, and e ａ is the position control error within the actuator coordinates, and p １２ is the second constant, and f ｆｆａ is the feedforward control signal within the actuator coordinates, The control method according to any one of claims 1 to 3.

5. The position control error in actuator coordinates is determined as follows: e ａ = T ｅ .e In the above equation, T ｅ is a position conversion matrix from the object position coordinates to the actuator coordinates, and e is the position control error. The control method according to claim 4.

6. The feedforward control signal in actuator coordinates is determined as follows: f ｆｆａ = T ｆｇｃ . f ｆｆ , In the above equation, T ｆｇｃ is the transformation matrix from the feedforward force in the control coordinates to the actuator coordinates, and f ｆｆ is the feedforward control signal. The control method according to claim 4 or 5.

7. Combining the feedback control signal and the feedforward control signal into the actuator input includes a decoupling step, where T ｆｇｃ corresponds to the decoupling step, and the control method according to claim 6.

8. Determining the feedback control signal is based on a proportional control constant and a derivative control constant, and p １１ is substantially equal to the proportional control constant, and p １２ is substantially equal to the derivative control constant, the control method according to any one of claims 4 to 7.

9. The method further includes: reconstructing an actual corrected actuator input based on the actual position of the object; and estimating the actuator gain correction based on the difference between the actuator input and the corrected actuator input. The control method according to any one of claims 1 or 2.

10. The control method according to any one of claims 1 to 9, wherein the method is arranged to control the position of the object in six degrees of freedom.

11. The control method according to any one of claims 1 to 10, wherein the object is an optical element of a projection system of a lithographic apparatus.

12. A control system for controlling the position of an object, the control system comprising: an electromagnetic actuator arranged to exert an actuator force on the object based on an actuator input; a feedback control device arranged to provide a feedback control signal based on a position control error between a desired position and an actual position of the object; a feedforward control device arranged to provide a feedforward control signal based on the desired position; comprising the control system being arranged to combine the feedback control signal and the feedforward control signal into an actuator input; the control system comprising an actuator gain correction device, the actuator gain correction device being arranged to determine an actuator gain correction based on the actuator input and the actual position of the object, and to apply the actuator gain correction to the actuator input to provide a corrected actuator input to the electromagnetic actuator; control system.

13. The control system according to claim 12, wherein the control system comprises a decoupling device for applying a decoupling step to the combined feedback control signal and feedforward control signal.

14. The control system according to claim 12 or 13, based on the feedforward control signal and the position control error or the feedback control signal.

15. The actuator gain correction device is arranged to determine the actuator gain correction as follows: K = -Γ∫(p １１ e ａ (t) + p １２ (de ａ (t)) / dt)f ｆｆａ (t)dt In the above equation, K is the actuator gain correction, and p １１ is the first constant, and e ａ is the position control error within the actuator coordinates, and p １２ is the second constant, and f ｆｆａ is the feedforward control signal within the actuator coordinates. The control system according to any one of claims 12 to 14.

16. The actuator gain correction device is arranged to determine the position error in actuator coordinates as follows: e ａ = T ｅ .e In the above equation, T ｅ is a position conversion matrix from the object position coordinates to the actuator coordinates, and e is the position control error. The control system according to claim 15.

17. The actuator gain correction device is arranged to determine the feedforward control signal in actuator coordinates as follows: f ｆｆａ = T ｆｇｃ .f ｆｆ , In the above equation, T ｆｇｃ is a force conversion matrix from the feedforward force in the control coordinates to the actuator coordinates, and F ｆｆ is the feedforward control signal. The control system according to claim 15 or 16.

18. Combining the feedback control signal and the feedforward control signal into the actuator input includes a decoupling step, where T ｆｇｃ corresponds to the decoupling step, the control system according to claim 17.

19. The feedback control device includes a proportional control constant and a derivative control constant, and p １１ is substantially equal to the proportional control constant, and p １２ is substantially equal to the derivative control constant, The control system according to any one of claims 15 to 18.

20. the actuator gain correction device is configured to reconstruct an actual corrected actuator input based on the actual position of the object, and estimate the actuator gain correction based on the difference between the actuator input and the corrected actuator input, the control system according to claim 12 or 13. **Claim 21** The control system according to any one of claims 12 to 20, wherein the control system is arranged to control the position of the object in six degrees of freedom. **Claim 22** The control system according to any one of claims 12 to 21, wherein the control system comprises a position measurement system for determining the actual position of the object. **Claim 23** A lithographic apparatus comprising a control system as claimed in any one of claims 12 to 22 for controlling the position of an object of the lithographic apparatus. **Claim 24** The lithographic apparatus according to claim 23, wherein the object is an optical element of the lithographic apparatus. **Claim 25** The lithographic apparatus according to claim 23 or 24, wherein the object is a mirror of a projection system of the lithographic apparatus.

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