JP2011233002A - Disturbance observer, feedback compensator, positioning device, exposure device and method for designing disturbance observer - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a disturbance observer, a feedback compensator, a positioning device, exposure device and a method for designing the disturbance observer which can stabilize a resonance phase using a phase lead compensator and a phase lag compensator which are based on a transfer function decided by a designer without a process of trial and error.SOLUTION: As for a disturbance observer 21 for estimating disturbance d which is given to an control target 28, its complementary sensitivity function parameters are designed based on a disturbance model including a resonance element of the control target 28 as a part of the disturbance d.

Description

  The present invention relates to a disturbance observer and a feedback compensator, and relates to a positioning apparatus, an exposure apparatus, and a disturbance observer design method including at least one of them.

  In a positioning control system such as a magnetic disk device, there is disclosed a method of selecting a method for stabilizing each resonance when there are a plurality of mechanical resonances (see Patent Document 1 and Non-Patent Document 1). Further, Patent Document 2 discloses a technique for stabilizing all resonance modes of a multi-inertia resonance system by applying a disturbance observer and resonance ratio control. Non-Patent Documents 2 and 3 disclose disturbance observers that estimate a disturbance input to a control target such as a positioning control device.

JP 2000-254256 A JP 2007-43884 A

Takenori Atsumi, Toshiro Arisaka, Toshihiko Shimizu and Takashi Yamaguchi, “Vibration Servo Control Design for Mechanical Resonant Modes of a Hard-Disk-Drive Actuator”, JSME International Journal Onishi, “New servo technology in mechatronics”, IEEJ Transactions D, Vol. 107, no. 1, pp. 83-pp. 86 (1987) Akira Shimada, “Motion Control”, Ohmsha, p. 157-164

  By the way, in the positioning control method disclosed in Patent Document 1, it is necessary to determine the transfer function (number of phase compensators, parameters, etc.) by trial and error while the designer confirms the phase characteristics of the round transfer function. There was a problem.

  In the positioning control method disclosed in Patent Document 2, a semi-closed control system is assumed. In a semi-closed control system (control using position information on the motor side), since the mode influence constants of resonance included in the controlled object are all positive, a control device equipped with a disturbance observer or feedback compensator is used for phase advance compensation. The phase of resonance can be stabilized by the device.

  On the other hand, in an exposure apparatus and the like and an HDD (Hard Disk Drive) disclosed in Patent Document 1, a fully closed control system (control using position information on the load side) is assumed. However, in the fully closed loop control system, the resonance mode influence constant varies between positive and negative, and therefore, in order to stabilize the resonance phase, a control device equipped with a disturbance observer or feedback compensator requires a large number of phase advance compensators and many There was a problem that it was necessary to combine with a phase lag compensator.

  The present invention has been made in view of the above-described points, and in a fully closed control system, a phase lead compensator and a phase lag compensator based on a transfer function determined by a designer without trial and error are used. It is an object of the present invention to provide a disturbance observer, a feedback compensator, a positioning device, an exposure apparatus, and a disturbance observer design method.

  The present invention has been made in order to solve the above-described problem, and in a disturbance observer for estimating a disturbance applied to a controlled object, the present invention is based on a disturbance model including the resonance component of the controlled object as a part of the disturbance. The disturbance observer is characterized in that the parameters of the complementary sensitivity function are designed.

  Further, according to the present invention, the pole of the disturbance observer is a straight line parallel to the imaginary axis in the complex plane, and the origin is defined by a first straight line having a negative real component that is the intersection of the straight line and the real axis. A first region surrounded by a circle having a radius larger than the distance between the first straight line and the imaginary axis, and two straight lines that pass through the origin and are not parallel to the imaginary axis among the circles at the center. A disturbance observer characterized in that the disturbance observer is arranged in a region common to the second region including the real axis among the regions separated by the second and third straight lines that are symmetric with respect to the real axis. is there.

  The disturbance observer according to the present invention is characterized in that the pole is arranged in a region on a complex plane so that an H∞ norm of the complementary sensitivity function of the disturbance observer is minimized.

  The present invention further includes a disturbance observer and a subtracter that subtracts an observation signal indicating the position of the control target from a signal indicating the target trajectory of the control target and outputs the subtraction result to the disturbance observer. This is a feedback compensator.

  In addition, the present invention is a positioning device including a feedback compensator that controls movement of the controlled object, and a stage device that includes a drive unit that moves the controlled object according to the control.

  The present invention also provides an exposure apparatus that exposes a mask pattern held on a mask stage onto a photosensitive substrate held on a substrate stage, wherein a positioning device is used as at least one of the mask stage and the substrate stage. An exposure apparatus is provided.

  Further, the present invention is a design method for designing a disturbance observer for estimating a disturbance applied to a controlled object, wherein the parameter of the complementary sensitivity function is based on a disturbance model that includes the resonance component of the controlled object as part of the disturbance. Is a disturbance observer design method characterized by having a process of designing.

  According to the present invention, in the disturbance observer, the parameters of the complementary sensitivity function are designed based on the disturbance model that includes the resonance component to be controlled as part of the disturbance. Thereby, the disturbance observer can stabilize the phase of resonance by the phase advance compensator and the phase lag compensator in the fully closed control.

  The disturbance observer has poles arranged in a region on the complex plane so that the H∞ norm (peak value) of the complementary sensitivity function of the disturbance observer is minimized. Thus, the designer can design a disturbance observer that stabilizes the phase of resonance by the phase advance compensator and the phase lag compensator in the fully closed control without trial and error of the transfer function.

  The feedback compensator includes a disturbance observer and a subtracter that subtracts an observation signal indicating the position of the control target from a signal indicating the target trajectory of the control target and outputs the subtraction result to the disturbance observer. Thereby, the feedback compensator can stabilize the phase of resonance by the phase lead compensator and the phase lag compensator in the fully closed control. The feedback compensator can be positioned with high accuracy by stabilizing the phase of resonance.

It is a block diagram of the stage apparatus in embodiment of this invention. It is a block diagram of a control device which performs positioning (vibration suppression) control of a plate holder, and an expansion system. It is a figure which shows the robust servo system (positioning servo system) containing a subtractor, an expansion system, and a feedback compensator by transfer function expression. It is a figure which shows the area | region which arrange | positions a pole. FIG. 5 is a Bode diagram showing frequency characteristics from a control input to a table position in a robust servo system in which a plate stage is controlled. It is a table | surface which shows the example of the parameter identified from the plate stage. 6 is a table showing parameters in design examples 2 and 3. It is a Bode diagram which shows the frequency characteristic of the feedback compensator in the design examples 1-3. It is a figure which shows the complementary sensitivity function T (s) in the design examples 1-3. It is a figure which shows the sensitivity function S (s) in the design examples 1-3. It is a figure which shows the disturbance suppression characteristic in the design examples 1-3. It is a simulation result of target value step response. It is a simulation result of a step disturbance response. It is a block diagram of the exposure apparatus to which the stage apparatus is applied.

  Embodiments of the present invention will be described in detail with reference to the drawings. FIG. 1 shows the configuration of a stage device (positioning device) in an embodiment of the present invention.

  Two X-axis guides 2 extending in the X-axis direction are laid in parallel on the upper surface of the base 1 at intervals in the Y-axis direction (one of them is shown in the figure). An X carriage 3 is movably provided so as to straddle both sides and the upper part of each X-axis guide 2. A Y beam guide 4 extends in the Y-axis direction (direction perpendicular to the paper surface) on the top of the X carriage 3, and is suspended and fixed in a bridge shape connecting the two X carriages 3.

  A plurality of air bearings 5 are arranged between the X carriage 3 and the X guide 2. The air bearing 5 is fixed to the X carriage 3, and the X carriage 3 and the Y beam guide 4 that are supported floating (non-contact) with respect to the X guide 2 are guided by the X guide 2 and are movable in the X-axis direction. It becomes the composition of.

  A Y carriage 7 is placed on the Y beam guide 4. A plurality of air bearings 8 (base bearings) are arranged between the Y carriage 7 and the upper surface of the Y beam guide 4, and between the Y carriage 7 and both side surfaces of the Y beam guide 4, the air bearing 9. A plurality of (side bearings) are arranged. The air bearings 8 and 9 are fixed to the Y carriage 7, and the Y carriage 7 that is floating (non-contact) supported by the Y beam guide 4 is guided by the Y beam guide 4 and is movable in the Y-axis direction. It has become.

  On the Y carriage 7, the plate table 10 is supported by a plurality of support mechanisms (not shown). On the plate table 10, a plate holder 11 for sucking and holding a glass substrate and the like, and a position sensor 12 for measuring the position are provided. The position sensor 12 (measurement target unit) is, for example, a movable mirror that forms part of an interference measuring instrument (laser interferometer), and laser light from an external light source is used as measurement light on the movable mirror. The position of the plate holder 11 is precisely measured by irradiating and reflecting and causing interference with the reference light in the interference measuring instrument.

  An actuator (drive unit, for example, X linear motor) (not shown) is provided at both ends (the front side and the back side of the paper surface) of the Y beam guide 4 and is mounted on the X carriage 3 by driving the X linear motor. The object moves along the X guide 2 in the X axis direction. A Y linear motor 6 is provided on the side surface of the Y carriage 7, and the structure mounted on the Y carriage 7 is moved in the Y axis direction along the Y beam guide 4 by driving the Y linear motor 6. . In other words, the X stage 100 is formed by the X carriage 3, the Y beam guide 4, and the X linear motor. The X stage 100 moves in the X axis direction, and the Y carriage 7, the plate table 10, the plate holder 11, and the position The stage 12 is configured such that the sensor 12 and the Y linear motor 6 form a Y stage 200 and the Y stage 200 moves in the Y-axis direction.

Next, the disturbance observer will be described. Hereinafter, as an example, the plate holder 11 is a control target.
FIG. 2 is a block diagram of a control device that performs positioning (vibration suppression) control of the plate holder 11 and an enlargement system in one embodiment. In this block diagram, the control object (plate holder 11) is represented as a control object 28 included in the enlargement system 26. This control device includes a disturbance observer 21. In FIG. 2, the disturbance observer 21 and the expansion system 26 are shown by transfer function expressions.

  In FIG. 2, a control input u is an input for positioning control of the position of the plate holder 11 that is a control target (the control target 28 included in the enlargement system 26 in FIG. 2).

  Hereinafter, in the present specification, the estimated value of the disturbance d is denoted by d ^. In addition, when a hat symbol is attached on a character in the mathematical expression, an estimated value of the physical quantity indicated by the character is indicated.

Subtractor 20, an estimate d ^ of the disturbance d to the disturbance observer 21 is calculated, subtracted from the nominal control input u _n, as obtained control value input u, and outputs the control target and the filter 24.

  The expansion system 26 is a model to be controlled in consideration of noise v and disturbance d. Here, u is a control input, and y is an observation signal.

  The control object 28 multiplies the addition result input from the adder 27 by the first transfer function P (s), which is the transfer function of the control object 28, and outputs the multiplication result to the adder 29. For example, the multiplication result output by the control object 28 is a value related to the position of the plate holder 11 (see FIG. 1) that moves according to the addition result input from the adder 27.

  White noise v is input to the adder 29 as an example of noise. The adder 29 adds the multiplication result input from the control object 28 and the white noise v, and outputs the addition result (observation signal) y to the disturbance observer 21. The observation signal y is, for example, position information of the plate holder 11 obtained by the position sensor 12 (see FIG. 1).

  The disturbance observer 21 includes a filter 23, a filter 24, and a subtracter 25. The filter 23 is a filter represented by the product of the transfer function Q (s) and the inverse system of the transfer function P (s). The observation signal y is input from the adder 29 to the filter 23. The filter 23 multiplies the input observation signal y by the product of the transfer function Q (s) and the inverse system of the transfer function P (s), and outputs the multiplication result to the subtractor 25.

The filter 24 is a filter represented by a transfer function Q (s). A control input u is input to the filter 24. The filter 24 multiplies the control input u by the transfer function Q (s) and outputs the multiplication result to the subtracter 25.
The subtracter 25 subtracts the output (multiplication result) of the filter 24 from the output (multiplication result) of the filter 23, and outputs (feeds back) it to the subtracter 20 as an estimated value d ^ of the disturbance d.

Next, the design of the disturbance observer 21 will be described.
The controlled object 28 is a multi-inertia system represented by the formula (1). Here, K _{p0 is} the gain of the rigid body mode, K _{pi is} the gain of the resonance mode, ω _{pi is the} resonance frequency, ζ _{pi is the} damping coefficient, and np is the number of resonance modes in the controlled object 28.

  Moreover, Formula (1) can be converted into Formula (2) as a state space model of the control object 28.

  A model of disturbance d (hereinafter referred to as “disturbance model”) is expressed by Expression (3).

The third term of Equation (3) represents the transfer function of the resonance model to be controlled. Where ω _{di is the} periodic disturbance frequency, ζ _{di is} the attenuation coefficient of the periodic disturbance model, nd is the number of periodic disturbance models, A _{i is the} value 1 for the resonance mode to be stabilized, and 0 otherwise. Coefficient, m: type of feedback control system (m> 0) obtained by equivalent deformation. If the disturbance observer 21 is m ≧ 1, the disturbance observer 21 alone can suppress the disturbance even if U _n = 0. Further, the feedback compensator obtained by equivalently modifying the disturbance observer 21 can suppress step disturbance even if U _n = 0 if m ≧ 1. Therefore, high integration characteristics can be obtained by increasing the type m of the control system.

  The disturbance observer 21 shown in FIG. 2 exemplifies a case where the disturbance model is expressed as a multi-inertia model, but the way of expressing the transfer function of the disturbance model is particularly limited to Expression (3). is not.

  Moreover, Formula (3) can be converted into Formula (4) as a state space model of the disturbance d. Here, v is noise (white noise).

  The disturbance observer 21 can suppress the disturbance d applied to the controlled object by the third term of the equation (3) indicating the disturbance model, and can stabilize the resonance mode specified by the third term of the equation (3). . Therefore, the resonance model (resonance component) of the disturbance indicated by the third term of Equation (3) includes the resonance model (resonance component) of the control target indicated by the second term of Equation (1). Then, by combining Expressions (2) and (4), Expression (5) indicating the state space model of the expansion system 26 is obtained.

  Further, the observer gain L of the disturbance observer 21 is added to the expression (5) to obtain an expression (6) indicating a state space model of the disturbance observer 21.

The pole of the disturbance model (such as the resonance mode of the controlled object 28) appears as the zero point of the sensitivity function S (s) = 1−Q (s) of the disturbance observer 21. Therefore, the peak of the sensitivity function is compressed at the resonance mode frequency specified by the third term of Equation (3).
The fact that the disturbance observer 21 expressed in the transfer function shown in FIG. 2 is equivalent to the equation (6) is described in Non-Patent Document 3 Section 7-2 (p.157-164).

  In the state space model of the disturbance observer 21 shown in Expression (6), the transfer function from the control input u output by the subtractor 20 to the disturbance estimated value d ^ is -Q (s). Here, the transfer function Q (s) and the complementary sensitivity function T (s) of the disturbance observer 21 have a relationship of T (s) = Q (s).

  If the poles of the disturbance observer 21 shown in Expression (6) are appropriately arranged on the complex plane, the disturbance observer 21 can suppress the disturbance d and stabilize the designated resonance mode.

  FIG. 3 shows a positioning servo system including a subtractor 40, an enlargement system 26, and a feedback compensator (control device) 30. The subtractor 40 subtracts the observation signal y from the target trajectory r and outputs the subtraction result to the feedback compensator 30.

  The feedback compensator 30 acquires the subtraction result output from the subtractor 40, multiplies the transfer function C (s), and outputs the multiplication result to the expansion system 26. Here, when Un = 0, the transfer function C (s) of the feedback compensator 30 is expressed by the following equation by equivalently modifying the transfer functions of the adder 20 and the disturbance observer 21 (in the solid line frame shown in FIG. 2). 7).

Next, the pole arrangement of the control system will be described.
When the pole of the equation (6) indicating the state space model of the disturbance observer 21 is arranged in the region on the complex plane, the zero point of the complementary sensitivity function T (s) of the disturbance observer 21 moves according to the arrangement of the pole. . Depending on the position of this zero point, the H∞ norm (peak value) of the complementary sensitivity function T (s) (= Q (s)) and the sensitivity function S (s) (= 1−Q (s)) becomes very large. . When these H∞ norms are large, the stability margin decreases and the positioning performance deteriorates.

  Therefore, the pole of the equation (6) indicating the state space model of the disturbance observer 21 is set on the complex plane so that the H∞ norm of the complementary sensitivity function T (s) and the sensitivity function S (s) is as small as possible. Place in the stable area. Generally, if the H∞ norm of the complementary sensitivity function T (s) is reduced, the H∞ norm of the sensitivity function S (s) is also reduced at the same time. Therefore, the complementary sensitivity function T (s) will be considered below. Arrange the poles.

  In the following, as an example, optimal pole placement is performed using a linear matrix inequality (LMI) so that the complementary sensitivity function of the disturbance observer is minimized. First, when the transfer characteristic from the control input u to the disturbance estimated value d ^ shown in Expression (6) is expressed by a state equation, Expression (8) is obtained. This transfer characteristic is -T (s) =-Q (s). Moreover, the dual system of Formula (8) is represented by Formula (9).

  FIG. 4 shows a region 44 (shaded portion) where a pole is arranged. The region 44 where the pole is disposed is a common region common to the alpha (α) stable region, the circular stable region, and the conic sector region.

  These areas will be described. First, the alpha stable region is a straight line parallel to the imaginary axis (Im), and from the straight line 40 where the real component α that is the intersection of the straight line and the real axis (Re) has a negative value, the negative direction of the real axis. It is an area that extends to That is, the real component α of the straight line 40 is a negative value (α <0). Further, α and the radius r of the circle have a relationship of r> | α |.

  The circle stable region is a region surrounded by a circle 41 having a radius r larger than the distance between the straight line 40 and the imaginary axis, out of circles centered on the origin. The conic sector region is a conic region including the real axis among the two straight lines that pass through the origin and are not parallel to the imaginary axis, and are separated by the straight lines 42 and 43 that are symmetrical with respect to the real axis. . That is, the angle θ [rad] formed by the straight lines 42 and 43 is smaller than the angle π [rad].

  Next, a feedback gain K for arranging the poles in the region 44 is obtained. The LMI that makes the H∞ norm of the complementary sensitivity function T (s) and the sensitivity function S (s) as small as possible is expressed by Equation (10).

  In addition, the LMI that arranges the poles in the region 44 is expressed by Expression (11) using the Kronecker product.

By obtaining the solutions X ^* and Y ^* that simultaneously satisfy the equations (10) and (11), it is possible to obtain the feedback gain K that makes the H∞ norm as small as possible and places the pole in the region 44. It is. The feedback gain K obtained in this way is expressed by Expression (12). Further, the optimum value of the observer gain L of the disturbance observer 21 is expressed by Expression (13) using the feedback gain K obtained by Expression (12). Then, from the optimum value of the observer gain L, the complementary sensitivity function Q (s) in the disturbance observer 21 can be obtained by Expression (8) or the like.

  Further, the transfer function of the disturbance observer 21 (see FIG. 2) and the transfer function C (s) of the feedback compensator 30 (see FIG. 3) can be determined from the complementary sensitivity function Q (s).

  As described above, in the disturbance observer 21 for estimating the disturbance d applied to the controlled object 28, the disturbance observer 21 is designed based on the disturbance model including the resonance component of the controlled object 28 as a part of the disturbance d. Features. Thereby, the disturbance observer 21 can stabilize the phase of resonance by the phase advance compensator and the phase lag compensator in the fully closed control.

  The pole of the disturbance observer 21 is a straight line parallel to the imaginary axis in the complex plane, and a straight line 40 having a negative real value component that is the intersection of the straight line and the real axis, and a circle centered on the origin. Of the first region surrounded by the circle 41 having a radius r larger than the distance between the straight line 40 and the imaginary axis, and two straight lines that pass through the origin and are not parallel to the imaginary axis, Of the regions delimited by straight lines 42 and 43 that are symmetrical to each other, the region 44 is arranged in a region 44 that is common to the second region including the real axis.

  The disturbance observer 21 is characterized in that poles are arranged in a region on the complex plane so that the H∞ norm of the complementary sensitivity function of the disturbance observer 21 is minimized. Thus, the designer designs the disturbance observer 21 that stabilizes the phase of resonance by the phase advance compensator in the fully closed control without trial and error of the transfer function (number of phase compensators and parameters). Can do.

  Further, the feedback compensator 30 subtracts the observation signal y indicating the position of the control target 28 from the disturbance observer 21 and the signal indicating the target trajectory r of the control target 28 and outputs the subtraction result to the disturbance observer 21. And 40. Thereby, the feedback compensator 30 can stabilize the phase of resonance by the phase lead compensator in the fully closed control. The feedback compensator 30 can be positioned with high accuracy by stabilizing the phase of resonance.

  In addition, the positioning device includes a feedback compensator 30 that controls the movement of the control target 28 and a stage device that includes a drive unit that moves the control target 28 according to the control.

  The disturbance observer 21 is a design method for estimating the disturbance d applied to the controlled object 28. The disturbance observer design method is based on a disturbance model including the resonance component of the controlled object 28 as a part of the disturbance d. It is characterized by having a designed process.

<Design example>
A design example of a control device included in the positioning device will be described. Here, the positioning device positions the table position (X2) of the plate stage (X axis) included in the stage device of the liquid crystal exposure apparatus shown in FIG. Further, the control system is configured as shown in FIG.

  FIG. 5 is a Bode diagram showing frequency characteristics from the control input to the table speed in the plate stage. Here, “before” and “after” indicate measured values of frequency characteristics in the plate stage. “Model” indicates the theoretical value of the frequency characteristic from the control input to the table speed in the four-inertia system model of the plate stage. Here, the transfer function P (s) of the four-inertia system model of the plate stage up to the table position is expressed by Expression (14).

FIG. 6 shows an example of parameters identified from the plate stage. According to the identified parameters, the absolute value of the mode constant (gain) K _p1 of the primary resonance is smaller than the absolute values of the mode constants (gains) K _p2 and K _p3 of the secondary and tertiary resonances. Further, the primary resonance frequency hardly changes.

Thus, in the control system design example proposed in this specification (hereinafter referred to as “design example 3”), the primary resonance is suppressed by the notch filter. The secondary and tertiary resonances are stabilized by phase compensation. Disturbance model D _promoted (s) in design example 3 is shown in equation (15).

  In Design Example 3 in consideration of resonance, the second term of the transfer function to be controlled shown in Expression (14) is added to the disturbance model shown in Expression (15). In the disturbance model of Equation (15), periodic disturbance is not taken into consideration, and thus the term corresponding to the second term of Equation (3) is omitted.

Further, for comparison with the design example 3, a disturbance model D _notch (s) in the design example 2 in which the first to third resonances are suppressed by the notch filter is shown in an equation (16). And in the design example 2 which considered resonance, the transfer function of the control object shown to Formula (14) is added to the disturbance model shown to Formula (16).

  On the other hand, in order to compare with the design examples 2 and 3, the resonance is not considered, and the model to be controlled is a rigid body model that ignores the second term of the equation (14) and is designed using the same disturbance model as the equation (16). Hereinafter, the design example of the control system is referred to as “design example 1”.

FIG. 7 shows parameters in design examples 2 and 3. Here, the distance α, the radius r, and the angle θ that define the region in which the pole is arranged in the complex plane shown in FIG. 4 are determined as shown in (i) to (iii) below as an example. And
(I) The slowest pole (control band) determined by the distance α is about half the frequency of the primary resonance. This is because if the slowest pole is set to the primary resonance frequency or higher, the H∞ norm (peak) of the sensitivity function becomes a large value, and the feedback compensator has a very large gain in the high frequency region. In addition, if the slowest pole is too close to 0, the control system has a very low control bandwidth.
(Ii) The earliest pole determined by the radius r is approximately the same as the frequency of the tertiary resonance. This is because if the earliest pole is set to a predetermined frequency or higher, the feedback compensator has a very large gain in the high frequency region.
(Iii) θ = about 0.8π [rad]. This is because if the angle θ is reduced, the H∞ norm (peak) of the sensitivity function will increase.

  FIG. 8 is a Bode diagram showing frequency characteristics of the feedback compensator 30 (see FIG. 3) in Design Examples 1 to 3. In design example 1 (rigid), only phase lead compensation that stabilizes the rigid body mode is formed. In design example 2 (notch), in addition to the phase lead compensator that stabilizes the rigid body, a notch filter is formed in the vicinity of the first to third resonance frequencies. On the other hand, in design example 3 (proposed), a notch filter is formed only in the vicinity of the primary resonance frequency, and a phase delay compensator that stabilizes the resonance mode is formed for the secondary and tertiary resonances.

  FIG. 9 shows complementary sensitivity functions T (s) in design examples 1 to 3. FIG. 10 shows sensitivity functions S (s) in design examples 1 to 3. As shown in FIG. 10, in the design example 3, the gain in the vicinity of the secondary and tertiary resonance frequencies (portions indicated by ellipses in FIG. 10) is lower than that in the design example 2. It can be seen that the effects of the secondary and tertiary resonances are eliminated. On the other hand, in Design Example 1, it can be seen that the band is not widened according to the low frequency characteristics of the sensitivity function. Moreover, in the design example 1, since the primary resonance is not considered, it can be seen that the gain of the sensitivity function S (s) increases in the vicinity of 18 [Hz].

  FIG. 11 shows disturbance suppression characteristics in design examples 1 to 3. In design example 3, as compared with design example 1 and design example 2, the sensitivity function near the resonance frequency (the portion indicated by an ellipse in FIG. 11) is significantly compressed, and the control performance for suppressing disturbance is improved. It turns out that it is improving.

  FIG. 12 shows a simulation result of the target value step response. FIG. 13 shows the simulation result of the step disturbance response. As shown in FIG. 13, it can be seen that the residual vibration of the first to third resonances remains in the design method 2. On the other hand, in the design example 3, it can be seen that the residual vibrations of the secondary and tertiary resonances are removed.

Next, a stage apparatus for an exposure apparatus will be described.
The stage apparatus (positioning apparatus) according to the present embodiment described above can be used as a stage apparatus for an exposure apparatus that prints a fine circuit pattern on a glass substrate or a semiconductor substrate, for example.

  FIG. 14 is a block diagram of an exposure apparatus to which the above-described stage apparatus is applied. The exposure apparatus 31 includes an illumination optical system 32, a mask stage apparatus 33 that moves while holding the mask M, a projection optical system PL, and a substrate stage apparatus 35 that holds and moves the glass substrate P. Is done.

  The illumination optical system 32 includes a light source unit, a shutter, a secondary light source forming optical system, a beam splitter, a condenser lens system, a reticle blind, and an imaging lens system, all of which are not shown, and is held by a mask stage device 33. A predetermined illumination area (including a circuit pattern) on the mask M is illuminated with a uniform illuminance by the illumination light IL.

  The projection optical system PL is an optical system (for example, a refractive optical system) having a plurality of lens elements arranged at predetermined intervals along the direction of the optical axis AX, and the mask M is illuminated by illumination light IL from the illumination optical system 32. When the illumination area is illuminated, the illumination light that has passed through the mask M projects an upright image of a predetermined magnification of the circuit pattern of the illumination area on the mask M onto the glass substrate P via the projection optical system PL. Thus, the photoresist applied to the surface of the glass substrate P is exposed.

  The above-described stage apparatus of FIG. 1 is used for at least one of the mask stage apparatus 33 and the substrate stage apparatus 35. Here, when used as the mask stage device 33, the mask M is held by the plate holder 11, and when used as the substrate stage device 35, the glass substrate P is held by the plate holder 11.

  As described above, the exposure apparatus is an exposure apparatus that exposes the mask pattern held on the mask stage onto the photosensitive substrate held on the substrate stage, and the stage apparatus serves as at least one of the mask stage and the substrate stage. (Positioning device).

  The embodiment of the present invention has been described in detail with reference to the drawings. However, the specific configuration is not limited to this embodiment, and includes designs and the like that do not depart from the gist of the present invention.

  For example, an algorithm such as a simplex method or a genetic algorithm (GA) may be used to optimize the position where the pole is arranged.

  Note that a program for realizing the control system as described above may be recorded on a computer-readable recording medium, and the program may be read into a computer system and executed. Here, the “computer system” includes an OS and hardware such as peripheral devices. The “computer-readable recording medium” refers to a storage device such as a flexible medium, a magneto-optical disk, a portable medium such as a ROM and a CD-ROM, and a hard disk incorporated in a computer system. Further, the “computer-readable recording medium” refers to a volatile memory (RAM) in a computer system that becomes a server or a client when a program is transmitted via a network such as the Internet or a communication line such as a telephone line. In addition, those holding programs for a certain period of time are also included. The program may be transmitted from a computer system storing the program in a storage device or the like to another computer system via a transmission medium or by a transmission wave in the transmission medium. Here, the “transmission medium” for transmitting the program refers to a medium having a function of transmitting information, such as a network (communication network) such as the Internet or a communication line (communication line) such as a telephone line. The program may be for realizing a part of the functions described above. Furthermore, what can implement | achieve the function mentioned above in combination with the program already recorded on the computer system, what is called a difference file (difference program) may be sufficient.

DESCRIPTION OF SYMBOLS 1 ... Base 2 ... X-axis guide 3 ... X carriage 4 ... Y beam guide 5, 8, 9 ... Air bearing 6 ... Y linear motor 7 ... Y carriage 10 ... Plate table 11 ... Plate holder 12 ... Position sensor 20 ... Adder DESCRIPTION OF SYMBOLS 21 ... Disturbance observer 23 ... Filter 24 ... Filter 25 ... Subtractor 26 ... Expansion system 27 ... Adder 28 ... Control object 29 ... Adder 30 ... Feedback compensator 31 ... Exposure apparatus 32 ... Illumination optical system 33 ... Mask stage apparatus 35 ... Substrate stage device 40 ... Subtractor 100 ... X stage 200 ... Y stage

Claims (7)

In the disturbance observer that estimates the disturbance applied to the controlled object,
A disturbance observer characterized in that a parameter of a complementary sensitivity function is designed based on a disturbance model including the resonance component to be controlled as a part of the disturbance.   The pole of the disturbance observer according to claim 1 is a straight line parallel to the imaginary axis in the complex plane, the first line having a negative real value component at the intersection of the straight line and the real axis, and the origin A first region surrounded by a circle having a radius larger than the distance between the first straight line and the imaginary axis, and two straight lines that pass through the origin and are not parallel to the imaginary axis among the circles at the center. And the second region and the third straight line that are symmetrical to each other about the real axis are arranged in a region common to the second region including the real axis. Disturbance observer described in.   The disturbance observer according to claim 1 or 2, wherein the pole is arranged in a region on a complex plane so that an H∞ norm of the complementary sensitivity function of the disturbance observer is minimized. The disturbance observer according to any one of claims 1 to 3,
A subtractor for subtracting an observation signal indicating the position of the control target from a signal indicating the target trajectory of the control target, and outputting the subtraction result to the disturbance observer;
A feedback compensator comprising: The feedback compensator according to claim 4, which controls movement of the controlled object.
In accordance with the control, a stage device having a drive unit that moves the control target;
A positioning device comprising: An exposure apparatus that exposes a mask pattern held on a mask stage onto a photosensitive substrate held on a substrate stage,
An exposure apparatus comprising the positioning device according to claim 5 as at least one of the mask stage and the substrate stage. A design method for designing a disturbance observer for estimating a disturbance applied to a controlled object,
A disturbance observer design method comprising a step of designing a parameter of a complementary sensitivity function based on a disturbance model including the resonance component to be controlled as a part of the disturbance.

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