Systems and methods for position control

Abstract

The present disclosure provides a method for controlling an actuator, comprising providing a sequence of measurement samples at a first frequency, processing the sequence of measurement samples at the first frequency to provide at least one sequence of derivative measurement samples, downsampling the sequence of measurement samples and the at least one sequence of derivative measurement samples to a second frequency lower than the first frequency, transmitting the downsampled sequence of measurement samples and the at least one sequence of downsampled derivative measurement samples, using the downsampled sequence of measurement samples and the at least one sequence of derivative measurement samples to provide at least a sequence of control error samples, and using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency.

Description

[Technical Field]

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to EP application 23157050.8, filed February 16, 2023, the entire contents of which are incorporated herein by reference.

[0002]

[0002] The present invention relates to systems and methods for control, and in particular to position control of equipment, which can be used to control the position of equipment such as a wafer stage in a lithographic apparatus. [Background technology]

[0003] A lithographic apparatus is a machine configured to apply a desired pattern onto a substrate. Lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). A lithographic apparatus may, for example, project a pattern (often referred to as a "design layout" or "design") in a patterning device (e.g., mask) onto a layer of radiation-sensitive material (resist) provided on the substrate (e.g., wafer).

[0004] As semiconductor manufacturing processes continue to advance, the dimensions of circuit elements have continually decreased, while the number of functional elements, such as transistors, per device has steadily increased for decades, following a trend commonly referred to as "Moore's Law." To accommodate Moore's Law, the semiconductor industry pursues technologies that enable the creation of increasingly smaller features. To project a pattern onto a substrate, a lithography system may use electromagnetic radiation. The wavelength of this radiation determines the minimum size of the features patterned on the substrate. Typical wavelengths currently in use are 365 nm (i-line), 248 nm, 193 nm, and 13.5 nm. Lithography systems using extreme ultraviolet (EUV) radiation, with wavelengths in the 4 nm to 20 nm range, e.g., 6.7 nm or 13.5 nm, may be used to form smaller features on a substrate than lithography systems using radiation with a wavelength of, e.g., 193 nm.

[0005]

[0005] During the lithography process, the positions of various moving parts of the lithography apparatus are continuously measured. For example, the substrate is typically positioned on a movable wafer stage. To accurately overlay multiple patterns, it is essential that the position of the wafer stage is known as accurately as possible. This can ensure that the image is projected onto the substrate in the correct position.

[0006] Each moving part is typically provided with one or more actuators. These actuators are controlled by a controller that includes a control loop. The control loop typically compares the measured position with a position setting and makes corrections as needed. In its most general form, such a setting may appear relatively simple. However, in a lithography apparatus, the number of moving parts to be controlled and the relative velocities and accelerations of their relatively heavy components (e.g., wafer stages) generate a relatively large number of measurement data samples at a relatively high frequency. Other data, such as diagnostic data, may also need to be transferred over the same data link. Links for transferring data from one section of the machine to another, for example, from sensors to actuators, typically have limited bandwidth. Increasing the bandwidth may be an option, but it has limitations. For example, the cost of the data link is a limiting factor. On the other hand, new hardware is required, which may be expensive and bulky, requiring a redesign. In addition to speed, requirements for the data link may include stability and concurrency. The latter significantly impact the time and effort required to build a suitable data link, which may require several years of work. Also, during operation, relatively small changes in the operating equipment, such as the sample rate of sensors or actuators, can affect the data link, requiring recalibration. Therefore, data signals are typically downsampled before transmitting the data and then upsampled at the final destination. However, downsampling can have several drawbacks.

[0007] In a typical position control loop, control samples (e.g., for controlling an actuator) are generated at a downsampled frequency based on measurement samples that are typically acquired at a higher frequency, and therefore the control samples are updated at a lower frequency than the measurement frequency.

[0008] One of the most significant drawbacks of real-time upsampling is the delay of the effective upsampled signal. Since the latter is typically used to control an actuator, a delay in the control signal translates into a delay in the actuator response. In real-time position control, minimizing control delay is the goal. In state-of-the-art systems, even a microsecond reduction in delay is valuable. Reducing delay not only increases the bandwidth of the control loop, resulting in increased throughput (due to reduced settling time) but also improved accuracy. Accuracy can be improved by reducing the position control error or residual. In theory, several parameters can be controlled to reduce delay. For example, increasing the bandwidth of the data link can increase the frequency of the control samples. This would reduce the downsampling ratio and reduce the sample delay of the upsampled signal. However, due to the reasons mentioned above, e.g., the potential requirement to limit the frequency bandwidth in the controller to reject unwanted disturbances, the sample delay of the upsampled signal is practically the only controllable parameter. However, reducing this delay has proven extremely difficult.

[0009]

[0009] Various techniques are available to improve subsequent upsampling. One of them is predictive first-order hold. In this technique, a predictive sample derived based on the previous control sample may be applied between successive control samples. However, predictive first-order hold uses a downsampled low-frequency input, resulting in phase and gain errors.

[0010]

[0010] US2008140343A1 discloses an integrated digital cantilever controller. This controller comprises a post-processing unit 25, consisting of optional post-processing software and / or hardware, and a digital-to-analog (D / A) converter 83. The post-processing unit 25, among other things, adds a DC component 81 to the signal. Furthermore, since D / A converters typically operate at a sampling rate much higher than the sampling rate at which the drive signal is updated, the drive signal is held by a hold algorithm 82. The most common hold algorithms are zero-order and first-order holds, although higher-order hold algorithms are not excluded. The held signal is then converted to the analog domain using a D / A converter 83. The analog signal coming out of the D / A converter can be scaled using gain scaling 84 to a signal acceptable to the cantilever actuator.

[0011]

[0011] The relatively infrequent update of control samples introduces delay, which can usually be mitigated by using a zero-order hold or a predictive first-order hold scheme. In predictive schemes, the derivative is determined based on previous control samples, and therefore samples that are only available at a relatively low downsampling frequency. This inherently introduces delay, distortion, or both. An example of such a system is disclosed in US2008140343A1.

[0012]

[0012] US4094959 discloses a method for providing predictive capabilities in measuring a variable process parameter and controlling the process accordingly, utilizing a transformation of the process parameter measurement signal, which may be a time derivative of a delayed process parameter measurement signal of second or higher order. This transformation is combined with the output of a PI controller to which the process parameter measurement signal is provided as an input, thereby generating a process variable signal that is used, either directly or after passing through another controller, as a process control signal for controlling the process in response to the measured parameter. US4094959 also introduces delays in data processing, making it unsuitable for processes requiring real-time or at least faster data processing, such as position control of a relatively fast-moving object.

[0013]

[0013] The present disclosure aims to provide an improved system and method for position control that is capable of transferring and processing data in real time. Summary of the Invention

[0014]

[0014] The present disclosure provides a method for controlling an actuator, comprising: providing a sequence of measurement samples (x(t)) at a first frequency; processing the sequence of measurement samples (at a first frequency) to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); downsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a second frequency lower than the first frequency; transmitting a sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples; using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency; The present invention provides a method comprising:

[0015] In an embodiment, the step of processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples occurs substantially at the first frequency.

[0016] In an embodiment, the method includes processing a sequence of measurement samples at a first frequency to provide at least two sequences of derivative measurement samples.

[0017] In an embodiment, the step of providing at least a sequence of control error samples using the sequence of downsampled derivative measurement samples and at least one sequence of derivative measurement samples comprises: using the sequence of measurement samples to determine a sequence of control error samples at a second frequency; using at least one sequence of derivative measurement samples to determine at least one sequence of derivative control error samples at a second frequency; upsampling the sequence of control error samples and at least one sequence of derivative control error samples to a third frequency; and filtering the sequence of upsampled control error samples and at least one sequence of derivative control error samples to provide a sequence of actuator control samples; Includes.

[0018]

[0018] In one embodiment, the step of using the sequence of derivative measurement samples to determine at least one sequence of derivative control error samples includes a step of comparing the at least one sequence of derivative measurement samples with at least one sequence of derivative reference samples.

[0019] In an embodiment, the step of providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples comprises: upsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a third frequency greater than the second frequency; filtering the sequence of upsampled measurement samples and at least one sequence of derivative measurement samples to provide at least a sequence of upsampled measurement samples; and using the sequence of upsampled measurement samples to determine a sequence of control error samples; Includes:

[0020] In an embodiment, using the sequence of measurement samples to determine the sequence of control error samples includes comparing the sequence of measurement samples to a sequence of reference samples.

[0021] In some embodiments, the third frequency is in the range of 100 kHz to 5 MHz. The first frequency and the third frequency may be of the same order of magnitude. The second frequency may be an order of magnitude smaller than the first frequency and / or the third frequency.

[0022]

[0022] In one embodiment, the step of providing at least a sequence of control error samples using a sequence of downsampled measurement samples and at least one sequence of derivative measurement samples includes a step of upsampling and a step of using the sequence of upsampled samples in an nth-order hold filtering step, where n is an integer greater than or equal to 1.

[0023] According to another aspect, the present disclosure provides a system for controlling an actuator, comprising: a measurement section for providing a sequence of measurement samples (x(t)) at a first frequency and for processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); a downsampler for downsampling at least the sequence of measurement samples to a second frequency lower than the first frequency; a control section for transmitting a sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; processing means for using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples to provide at least a sequence of control error samples, and for using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency; To provide a system comprising:

[0024] In an embodiment, the processing means comprises a subtractor for comparing the input with at least one sequence of reference samples to provide a sequence of control error samples.

[0025]

[0025] In one embodiment, the processing means comprises an upsampler for upsampling at least one input sequence and providing at least one upsampled output sequence, and a filter for filtering the at least one upsampled output sequence to provide a sequence of actuator control samples.

[0026]

[0026] According to yet another aspect, the present disclosure provides a position control system provided with at least one system as described above.

[0027] According to another aspect, the present disclosure provides a lithographic apparatus provided with at least one system as described above. [Brief explanation of the drawings]

[0028]

[0028] Embodiments of the present invention will now be described, by way of example only, with reference to the accompanying schematic drawings, in which:

[0029] [Figure 1] 1 shows a schematic overview of a lithographic apparatus; [Figure 2] 1 shows an exemplary diagram of a conventional position control system. [Figure 3] 10 graphically illustrates the delay caused by a zero-order hold filter. [Figure 4] 1 graphically illustrates the delay introduced by a conventional predictive first-order hold filter; [Figures 5A-5C] 1A-1D show diagrams of respective embodiments of a system according to the present disclosure; [Figure 6] 1 shows a diagram illustrating a signal provided by a first-order hold using the method and system of the present disclosure. [Figure 7] 1 shows a diagram illustrating a signal provided by second-order hold-up sampling using the method and system of the present disclosure. [Figure 8A-8B] 10A and 10B show examples of gain and phase behavior of the disclosed method using first-order hold compared to conventional zero-order hold and predictive first-order hold, respectively. DETAILED DESCRIPTION OF THE INVENTION

[0030]

[0029] In this specification, the terms "radiation" and "beam" are used to encompass all types of electromagnetic radiation, including ultraviolet (e.g., wavelengths of 365, 248, 193, 157, or 126 nm) and EUV (extreme ultraviolet, e.g., wavelengths in the range of about 5 to 100 nm).

[0031]

[0030] As used herein, the terms "reticle," "mask," or "patterning device" may be broadly interpreted to refer to a general patterning device that can be used to impart an incident radiation beam with a patterned cross-section that corresponds to the pattern to be created on a target portion of a substrate. The term "light valve" may also be used in this context. In addition to standard masks (transmissive or reflective, binary, phase-shifting, hybrid, etc.), examples of other such patterning devices include programmable mirror arrays and programmable LCD arrays.

[0032]

[0031] As used herein, the terms "upsampling" or "downsampling" relate to digital signal processing. Here, upsampling, extension, and interpolation are terms related to the resampling process in multi-rate digital signal processing systems. Upsampling may be synonymous with extension, or may refer to the overall process of extension and filtering (interpolation). When upsampling is performed on a sequence of samples of a signal or other continuous function, an approximation of the sequence obtained by sampling the signal at a higher rate is obtained. Similarly, downsampling is the process of reducing the sample rate of a signal.

[0033] 1 schematically depicts a lithographic apparatus LA. The lithographic apparatus LA comprises an illumination system (also called an illuminator) IL configured to condition a radiation beam B (e.g. UV radiation, DUV radiation or EUV radiation), a mask support (e.g. a mask table) MT constructed to support a patterning device (e.g. a mask) MA and connected to a first positioner PM configured to accurately position the patterning device MA according to certain parameters, a substrate support (e.g. a wafer table) WT constructed to hold a substrate (e.g. a resist-coated wafer) W and connected to a second positioner PW configured to accurately position the substrate support according to certain parameters, and a projection system (e.g. a refractive projection lens system) PS configured to project a pattern imparted to the radiation beam B by the patterning device MA onto a target portion C (e.g. comprising one or more dies) of the substrate W.

[0034]

[0033] In operation, the illumination system IL receives a radiation beam from the radiation source SO, for example via the beam delivery system BD. The illumination system IL may include various types of optical components, such as refractive, reflective, magnetic, electromagnetic, electrostatic and / or other types of optical components, or any combination thereof, for directing, shaping and / or controlling the radiation. The illuminator IL may be used to condition the radiation beam B so that it has a desired spatial and angular intensity distribution in its cross-section in the plane of the patterning device MA.

[0035] The term "projection system" PS as used herein should be interpreted broadly to encompass various types of projection systems, including refractive optical systems, catadioptric optical systems, anamorphic optical systems, magnetic optical systems, electromagnetic optical systems and / or electrostatic optical systems, or any combination thereof, as appropriate to the exposure radiation used and / or other factors such as the use of an immersion liquid or the use of a vacuum. Any use of the term "projection lens" herein may be considered as synonymous with the more general term "projection system" PS.

[0036]

[0035] The lithographic apparatus LA is of a type in which at least a portion of the substrate may be covered with a liquid having a relatively high refractive index, such as water, to fill a space between the projection system PS and the substrate W, also referred to as immersion lithography. More information about immersion techniques is given in US 6,952,253, which is incorporated herein by reference.

[0037] The lithographic apparatus LA may also be of a type having two or more substrate supports WT (also called "dual stage"). In such a "multi-stage" machine, the substrate supports WT may be used in parallel, and / or preparation steps for a subsequent exposure of a substrate W may be performed on a substrate W located on one substrate support WT, while another substrate W on another substrate support WT is being used to expose a pattern onto the other substrate W.

[0038] In addition to the substrate support WT, the lithographic apparatus LA may comprise a measurement stage. The measurement stage is configured to hold a sensor and / or a cleaning device. The sensor may be configured to measure a property of the projection system PS or a property of the radiation beam B. The measurement stage may hold multiple sensors. The cleaning device may be configured to clean part of the lithographic apparatus, for example part of the projection system PS or part of a system for providing immersion liquid. The measurement stage may move below the projection system PS when the substrate support WT is spaced apart from the projection system PS.

[0039] In operation, the radiation beam B is incident on a patterning device, for example a mask MA, which is held on a mask support MT, and is patterned by a pattern (design layout) present on the patterning device MA. Having traversed the mask MA, the radiation beam B passes through the projection system PS, which focuses the beam onto a target portion C of the substrate W. Using the second positioner PW and the position measurement system IF, the substrate support WT may be accurately moved, for example to position various target portions C at focused and aligned positions in the path of the radiation beam B. Similarly, the first positioner PM, and possibly further position sensors (not explicitly depicted in FIG. 1 ), may be used to accurately position the patterning device MA with respect to the path of the radiation beam B. The patterning device MA and substrate W may be aligned using mask alignment marks M1, M2 and substrate alignment marks P1, P2. The illustrated substrate alignment marks P1, P2 occupy dedicated target portions, but may be located in spaces between the target portions. When the substrate alignment marks P1, P2 are located between target portions C, they are known as scribe-lane alignment marks.

[0040] To clarify the invention, a Cartesian coordinate system is used. The Cartesian coordinate system has three axes: x, y, and z. Each of the three axes is orthogonal to the other two. Rotation about the x-axis is referred to as Rx rotation. Rotation about the y-axis is referred to as Ry rotation. Rotation about the z-axis is referred to as Rz rotation. The x- and y-axes define a horizontal plane, to which the z-axis is perpendicular. The Cartesian coordinate system is not a limitation of the invention and is used merely for clarity. Alternatively, another coordinate system, such as a cylindrical coordinate system, may be used to clarify the invention. The orientation of the Cartesian coordinate system may be different, for example, so that the z-axis has a component along the horizontal plane.

[0041]

[0040] Referring generally to Figure 2, in a typical control loop, a measurement system including a measurement section 102 periodically generates measurement samples x(t). The measurement samples may be provided by a sensor 110. The measurement samples may be provided at a constant bit rate (sometimes referred to as a sample frequency, measurement frequency, or first frequency). Somewhere in the control loop, the measurement samples are generated based on a reference sample x(t). ref (t), is typically compared to a set of setpoint values, and the result is used to drive the actuator 126. As discussed in the introduction, the data link between the sensor and the actuator poses various challenges. Typically, the data link includes a control section 104. The controller forwards the measurement samples to the actuator section 106. Thus, in this disclosure, the control section 104 may represent the data link between the sensor section 102 and the actuator section 106.

[0042]

[0041] In equipment control, for example, wafer stage position control, it is generally desirable to have as little servo tracking error as possible. Part of the servo tracking error is caused by disturbances. For example, in the case of wafer stage position control, disturbances may come from air pressure fluctuations, forces exerted by connected cables and hoses, etc. Therefore, a certain minimum disturbance rejection of the control loop is generally sought. The greater the disturbance rejection, the less sensitive the servo tracking error is to disturbances.

[0043]

[0042] Disturbance rejection may be achieved by a feedback controller. In this case, the quantity of interest (i.e., the quantity to be controlled) is measured, and the difference between the measurement and the setpoint (i.e., the desired value of the quantity to be controlled) is the input to the feedback controller. This difference is called the "servo error." The feedback controller converts this "servo error" signal into an actuation signal that is sent to an actuator that can change the quantity to be controlled.

[0044]

[0043] In general, the so-called servo "bandwidth" frequency of the control loop has a significant impact on disturbance rejection. The exact relationship between servo bandwidth and disturbance rejection depends on the spectral content of the disturbance. However, in practice, it is said that a certain minimum servo bandwidth is sought to meet the required disturbance rejection. As will be described later, to optimize the data link at a certain bandwidth, it is preferable to limit the phase delay of the final actuator control signal relative to the original measurement signal.

[0045] The control loop may include, for example, a filter 112 and a downsampler 114. The controller includes a step of filtering the measurement samples to a reference value x ref A subtractor 116 may be provided for comparing the control sample i(t) with the control sample i(t) and outputting a difference or error value e(t). The controller may comprise a feedthrough controller 118 that provides the control sample i(t).

[0046]

[0045] To improve the subsequent upsampling result, a predictive first- or second-order hold scheme may be used. Here, the control section 104 may include a derivative estimator 120. The estimator may provide an estimate of the nth-order derivative of the control sample i(t), where the value of n may be an integer equal to or greater than 1, such as 1 or 2. The nth-order derivative of the control sample i(t) is estimated. The upsampling scheme of the control loop shown in Figure 2 may be referred to as a predictive nth-order hold.

[0047] The actuator section 106 may receive the value of the control sample i(t) and its optional nth-order derivative. The actuator section 106 may comprise an upsampler 122. The upsampler may increase the sample rate of the value i(t) and its optional nth-order derivative to a third frequency. Typically, the third frequency may be substantially the same as or of a similar magnitude to the first frequency or sample frequency. The actuator section 106 may further comprise one or more filters 124. The filters 124 may include anti-aliasing filters. In a practical embodiment, the upsampler 122 and the filters 124 may be integrated into a single unit. The actuator section 106 may comprise or control an actuator 126.

[0048] As mentioned in the introduction, in state-of-the-art complex machines such as lithography apparatuses, a vast number of sensor signals are fed to the controller 104. The controller, in turn, needs to control a large number of actuators. To enable the data link represented by the controller section 104 to accommodate the data stream, the sample rate is typically downsampled, for example, using a downsampler 114. In practice, the downsampling ratio may be significant to allow data transfer and controller compatibility. For example, in practice, measurement samples may be sampled at a sample rate or frequency greater than 100 kHz, for example, about 300-350 kHz. The data link of the controller 104 may process the measurement samples at a much lower sample rate or frequency, for example, 10 kHz, for example, less than 25 kHz. An upsampler may aim for a third frequency greater than 100 kHz, for example, about 300-350 kHz, to restore the measurement frequency.

[0049] Generally speaking, the first frequency or measurement frequency is typically equal to the second frequency or control frequency (f s,Mo The controller converts this measurement signal into a desired actuation signal i(t) (at a sample rate f s,Mo The actuation section 106 typically converts the second sample frequency f s,Mo Therefore, during most of the actuation section samples, no new "desired actuation" samples are received from the controller.

[0050]

[0049] Currently, there are several strategies commonly used to fill in these missing samples. It should be noted that acausal systems, such as the "Fundamental First-Order Hold (FOH)" method, cannot be used for the real-time signal processing required to control relatively fast-moving equipment in a lithography apparatus. An acausal system is a system that is not a causal system. Acausal systems depend on future input values, and possibly input values ​​from the past or present. This is in contrast to causal systems that depend only on present and / or past input values. For example, in audio or video processing, signal buffering allows the use of a causal upsampling system, allowing for near-perfect reconstruction of the original signal. However, for controlling moving equipment, the delays required for buffering are unacceptable, ruling out acausal systems.

[0051] Thus, in a predictive n-th order hold (FIG. 2), the control section 104 uses previous MO actuation samples to estimate the derivatives of the actuation signal. These derivatives are used to predict the actuation signal until the next MO sample. The drawback of using previous MO samples is that the MO operates at a relatively low second frequency, the control frequency. Therefore, older samples are required to estimate the derivatives. This has a negative impact on the phase behavior of the resulting actuation signal.

[0052]

[0051] Figures 3 and 4 illustrate the sample delay caused by conventional causal upsampling.

[0053] 3, the original measurement signal may be represented by a substantially continuous line 140. The upsampler 122 of the actuator section 106 may use a "zero-order hold" (ZOH) scheme, where the actuation value 142 calculated at time t is the sum of the values ​​over the entire time interval [t, t+T s,MO 〉, where T s,MO is the duration between successive samples (i.e., T s,MO[s]=1 / f s,MO [s -1 3, holding the sample values ​​results in an effective phase delay equivalent to Ts,MO / 2 (curve 144), where curve 144 is a representation of the effective upsampled or restored signal. The effective delay of the upsampled signal 144 relative to the original signal 140 is usually referred to as the "half sample delay" of the zero-order hold (ZOH) method.

[0054] Referring to FIG. 4, another approach uses so-called predictive first-order hold (predictive FOH). This uses the difference between the latest controller sample i(t) and the previous most recent controller sample i(t) to estimate the time derivative of the actuation signal. This time derivative is used as an input 146 to control the actuator. The predictive first-order hold may be incorporated into the estimator 120 of FIG. 2. As shown in FIG. 4, the effective estimated signal 148 has a smaller effective phase delay 150 relative to the original measurement signal 140 than the zero-order hold (FIG. 3). However, the predictive first-order hold introduces a gain error 152. The latter means that the effective control signal 148 may have a different amplitude than the measurement signal 140.

[0055] Another option called "delayed first order hold" can be used as it shows the causal relationship, however this option introduces a full sample delay and is therefore worse than ZOH in that sense.

[0056] In practice, designers often optionally increase the sampling frequency of the data link, including the control section 104, in order to reduce the "half-sample delay." However, as mentioned in the introduction, increasing the sample rate of the data link may not be feasible for various reasons. Given the huge amount of data and the costs of the associated equipment for processing the data, this inevitably increases the cost of the data processing equipment and the space required to locate this equipment. In state-of-the-art lithography apparatus, these costs and space often make this solution economically infeasible, or at least limiting. Increasing the sampling frequency increases the technical complexity, as calculations and communications must be performed faster.

[0057] 5A shows an embodiment of a system 100 for controlling an actuator according to the present disclosure. The measurement section 102 may include a filter 112. The filter converts a sensor sample x(t) from the sensor 110 into its nth derivative. For example, the first and / or second derivatives, i.e., x(t), x′(t), x″(t), etc. Note that a filter for deriving x(t) from its zeroth derivative, i.e., x(t) itself, is not necessarily an identity filter. The output x(t) may or may not be a filtered version of the filter's input x(t).

[0058] The first and higher order derivatives of the sensor samples may be generated substantially at a first frequency. The first sample frequency is substantially equal to the sample frequency of the measurement section 102. However, it may also be possible to generate the nth order derivatives of the measurement samples at a lower frequency, for example at a second frequency. Here, instead of downsampling the derivatives, the derivatives can be provided at a downsampled bit rate. Here, all measurement samples can be used to determine the nth order derivatives of the measurement samples, but the derivatives may be generated at a lower bit rate, typically at the frequency for downsampling. The latter saves computational power while retaining improved accuracy.

[0059] The filter 112 may be an anti-aliasing filter. An anti-aliasing filter (AAF) is a filter used before the downsampler 114. The filter 112 may be a low-pass filter for filtering the (derivative) measurement signal, for example, to reduce sensitivity to measurement noise. The filter 112 limits the bandwidth of the measurement signal to satisfy the Nyquist-Shannon sampling theorem across the entire band of interest. This theorem states that unambiguous reconstruction of a signal from samples is possible when the output of frequencies above the Nyquist frequency is zero. Therefore, a brick-wall filter would be an ideal AAF, but it is not practical. Practical AAFs trade off bandwidth reduction for increased aliasing. Practical anti-aliasing filters typically tolerate some aliasing and attenuate or distort in-band frequencies close to the Nyquist limit. For this reason, many practical systems perform oversampling, which is sampling higher than a perfect AAF would theoretically require, to ensure that all frequencies of interest can be reconstructed.

[0060] The filter 112 provides the filtered measurement signal samples and their n-th derivatives to the downsampler 114. The downsampler reduces the sample rate of the measurement signal samples to a second frequency that is lower than the first frequency. In practical embodiments, the second frequency may be significantly lower than the first frequency.

[0061] The downsampled samples are finally transferred via a data link to the actuator section where the samples are upsampled. Here, the downsampled samples may be provided to the control section 104. The control section may, in some embodiments, comprise processing means for providing control error samples. For example, the control section 104 may comprise a subtractor 116. The subtractor subtracts the measurement samples x(t) and their n-th derivatives (x'(t), x''(t), etc.) from a reference or setpoint value, i.e., x ref (t), x ref '(t), x ref The subtractor essentially compares the measurement signal and its derivative to a setpoint by subtracting one signal from the other. The subtractor outputs the difference between the measurement signal and a reference, essentially an error value e(t) and higher order derivatives of the error value e'(t), e''(t), etc.

[0062] The error values ​​provided by the subtractor may be provided to a feedthrough controller 118. A feedthrough controller may be essentially a controller that is in state space rather than frequency domain. The controller 118 may, for example, convert the control error samples e(t), e'(t), e"(t), etc. into transformed error samples i(t), i'(t), i"(t). The transformed error samples or actuator samples may represent values ​​suitable for controlling an actuator, such as current values ​​(amperes). Additional processing may be incorporated at various points before or after the controller 118, as illustrated with respect to Figures 5A-5C.

[0063] The result of the processing in the control section 104, all taking place at the second operating frequency of the processing board, is a sequence of actuator control signals i(t) and their nth order derivatives i′(t), i″(t), etc. These actuator control signals and derivatives are supplied to each actuator section 106 for each actuator. Now, as mentioned above, a typical state-of-the-art machine may have any number of actuators 126 and corresponding position sensors 110. However, for the sake of simplicity, the drawings and corresponding description refer to a single sensor and actuator and the data link therebetween.

[0064] The actuator section 106 may include an upsampler 122 that upsamples the signal provided by the controller 104 to a third frequency that is greater than the second frequency. In practical embodiments, the third frequency may be of the same order of magnitude as the first frequency. In practice, the third frequency, or actuation frequency, may be substantially the same as the measurement frequency or first frequency.

[0065] The actuator section 106 may typically comprise a filter 124 for filtering the upsampled actuator control signals and derivatives i(t), i'(t), i''(t). The filter 124 may be considered a reconstruction filter and may also be called an anti-imaging filter. The filter 124 may be used to reconstruct a smooth analog signal from a digital input, as in the case of a digital-to-analog converter (DAC) or other sampled data output device. Alternatively, the filter may provide a (highly) upsampled digital output. The reconstructed signal may, for example, comprise a sequence of upsampled actuator control samples.

[0066] In practical embodiments, upsampling and anti-imaging are not separate processing steps. Upsampling is usually performed using a specific anti-imaging filter (or reconstruction filter). Therefore, the upsampler 122 and the anti-imaging filter 124 may usually be integrated.

[0067] The reconstruction signal provided by the filter 124 may be provided to an actuator 126 that controls it.

[0068]

[0067] The sensor 110 may typically include a position sensor. The position sensor may include an interferometer. The sensor 110 may include a phase measurement device for measuring the phase difference between two interferometer light beams. The interferometer phase measurement device, or phase measurement board, converts the photodetector signals from the encoder and / or interferometer into fringe counts. Here, the phase measurement board may generate fringe count samples, typically at a sample frequency that is the same as or approximately the same as the first frequency of the measurement samples.

[0069] The actuator section 106, which drives the actuator 126, internally generates actuator samples i(t) at a sampling rate much higher than the data transfer frequency and, optionally, the control section frequency MO. The method of the present disclosure uses derivatives obtained from the fit by the anti-aliasing filter 112. These derivatives are downsampled, transferred, and used by the actuator section 106 to predict actuator samples between subsequent control samples, i.e., until the next control sample is available. See FIG. 5A.

[0070] 5B shows an embodiment in which the feedthrough controller 118 is located in the actuator section 106 and operates thereon (i.e., after the upsampler 122 and the anti-imaging filter 124). Here, the control section 104 generates control error samples and their derivatives. The latter are forwarded to the actuator section and upsampled. The control error signal transmitted at a second frequency and its derivative, also at the second frequency, are used to reconstruct the upsampled control error signal using a reconstruction filter / anti-imaging filter. The upsampled control error samples are provided to the feedthrough controller at a third frequency. The controller 118 converts the control error samples into actuator samples, or actuation signals, for controlling the actuator 126.

[0071] 5C, the data stream is centrally collected and distributed via a data link represented by section 104. A control function that compares a measurement sample value with a reference sample value and subsequently uses the difference, i.e., an error sample value, for actuator control may be located in actuator section 106. Actuator section 106 comprises an upsampler 122 and a filter 124. As mentioned above, upsampling and filtering may be combined into a single step, and upsampler 122 and filter 124 may actually be combined. A comparison step, for example, subtractor 116, may be located following filter 124. A controller block 118 may be located after the subtractor. Controller 118 may be connected to actuator 126.

[0072] 5C illustrates an embodiment in which the comparison with the setpoint, including, for example, subtraction of the signal sample value from the reference value in subtractor 116, is performed in actuator section 106. Here, the control function is substantially performed in actuator section 106, and thus after upsampler 122 and anti-imaging filter 124. Control section 104 shown in FIG. 5C essentially functions as a data transfer link at the downsampled data bit rate. As noted above, data transfer at the downsampled bit rate has various advantages over the alternative of transferring data at the higher bit rate of the measurement samples. In other words, transferring data at the same bit rate as the bit rate of the measurement samples presents several challenges that make it unattractive and / or economically unfeasible in practice, for example, due to increased capital and operating expenses.

[0073] In the embodiment of FIG. 5C, upsampler 122 (typically combined with filter 124) receives measurement samples x(t) and nth-order derivatives x′(t), x″(t), etc. of the measurement samples at a second frequency. Filter 124 provides an upsampled sequence of filtered measurement samples x(t). The filtered sequence x(t) is derived from the sequence of measurement samples x(t) and the sequence of nth-order derivative measurement samples x′(t), x″(t). The sequence of filtered measurement samples x(t) can be provided to subtractor 116. Subtractor 116 subtracts the sequence of filtered and upsampled measurement samples x(t) from the sequence of reference samples x ref5C , the upsampled measurement sample values ​​and reference sample values ​​provided to the subtractor 116 are typically in the digital domain.

[0074] It should be noted that the representation of each sample sequence, such as i(t), e(t), x(t), etc., may vary depending on its function in the control loop. The representation may depend on the function of each unit that provides each sequence of samples.

[0075]

[0074] The proposed solution is to apply a predictive n-th order hold scheme between two successive control signals, where the predictive n-th order hold scheme does not use the previous control sample but uses the value of the measurement sample and its derivative.

[0076]

[0075] Figure 6 shows an example of a first-order hold scheme according to the method and system of the present disclosure. Here, line 140 represents a continuous analog measurement signal. An ideal reconstructed measurement signal would be identical to the original measurement signal. The system of the present disclosure provides a sequence of actuator samples i(t) 160 in the digital domain. Represented in or converted to the analog domain, the system of the present disclosure provides an actuator control signal 162. Comparing the actuator control signal 162 to the actuator control signal 148 of Figure 4, the signal 162 using the first-order hold scheme according to the method and system of the present disclosure reduces phase error 164 and gain error 166.

[0077] 7 shows an example of a quadratic hold scheme in accordance with the disclosed method and system. Here, line 140 represents a continuous analog measurement signal. The disclosed system generates a sequence 170 of actuator control samples i(t). Represented in analog form, the actuator control samples become actuator control signal 172. As shown in FIG. 7, actuator control signal 172 has limited gain and phase errors and is relatively close to the original measurement signal 140.

[0078]

[0077] Figures 8A and 8B show the gain and phase behaviors, respectively, of a predictive first-order hold compared to a conventional ZOH and a first-order hold according to the disclosed method. In Figures 8A and 8B, the horizontal axis represents the frequency content of the measurement signal provided by the sensor. In Figure 8A, the vertical axis represents the gain error of the actuator control signal. In Figure 8B, the vertical axis represents the phase error (e.g., expressed in radians or rad) of the actuator control signal. In the figures, line 180 illustrates the behavior using a zero-order hold. The ZOH is the same for both the conventional system and the disclosed system. Line 182 represents the conventional system using a predictive first-order hold. Line 184 represents anti-aliasing filtering including a fit to the first derivative with a first-order hold using the disclosed system. Line 186 represents the conventional system using a predictive first-order hold and a fit to two derivatives. Line 188 represents the disclosed system using a first-order hold and a fit to the first and second derivatives.

[0079]

[0078] The first-order hold using the disclosed system has good phase behavior. Improved phase behavior is beneficial for achieving a sufficiently wide control loop bandwidth, as discussed above. Also, gain behavior may be significantly improved in bandwidths up to the threshold frequency 190. The disclosed system and method can significantly improve data transfer at downsampled bit rates in the control loop. This improvement is most noticeable at frequencies up to the threshold frequency.

[0080] An important advantage of the disclosed system and method relates to increasing the amount of data available to reconstruct the measurement signal. In the example shown, the amount of data is increased by a factor of 32 (10 kHz vs. 320 kHz). As a result, more parameters can be adjusted, making it possible to find an optimum between frequency behavior and reconstruction error (imperfect reconstruction of a sinusoidal input) on the one hand, and noise sensitivity (related to the amplification of noise between the input and output of the system) on the other hand.

[0081] The systems and methods of the present disclosure may be beneficial in any feedback loop where strict phase delay requirements apply, driving actuators as well as other real-time applications, which may include, but are not limited to, position control of wafer stages, reticle stages, electron beam deflection coils, optical elements such as lenses and mirrors, wafer handlers, mask handlers, control systems, lasers, precision motion control, automated manufacturing systems, etc.

[0082]

[0081] Although specific reference may be made in this text to the use of lithographic apparatus in the manufacture of ICs, it will be appreciated that the lithographic apparatus described herein have other applications, including 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, etc.

[0083] Although specific reference is made herein to embodiments of the invention in relation to lithography apparatus, embodiments of the invention may be used in other apparatus. Embodiments of the invention may form part of a mask inspection apparatus, a metrology apparatus, or any apparatus that measures or processes objects such as wafers (or other substrates) or masks (or other patterning devices). These apparatus may be generally referred to as lithography tools. Such lithography tools may use vacuum conditions or ambient (non-vacuum) conditions.

[0084]

[0083] Although specific reference has been made above to the use of embodiments of the present invention in relation to optical lithography, it will be understood that, where the context permits, the present invention is not limited to optical lithography and may also be used in other applications, for example imprint lithography.

[0085] Where the context permits, embodiments of the present invention may be implemented in hardware, firmware, software, or any combination thereof. Embodiments of the present invention may also be implemented as instructions stored on a machine-readable medium, which may be read and executed by one or more processors. A machine-readable medium may include any mechanism for storing or transmitting information in a form readable by a machine (e.g., a computing device). For example, machine-readable media may include read-only memory (ROM), random-access memory (RAM), magnetic storage media, optical storage media, flash memory devices, electrical, optical, acoustic, or other forms of propagated signals (e.g., carrier waves, infrared signals, digital signals, etc.), and others. Furthermore, firmware, software, routines, and instructions may be described herein as performing particular actions. However, it should be understood that such description is merely for convenience and that such actions actually result from a computing device, processor, controller, or other device executing the firmware, software, routines, instructions, etc., and, in doing so, may cause actuators or other devices to interact with the physical world.

[0086]

[0085] While specific embodiments of the present invention have been described above, it will be understood that the invention may be practiced otherwise than as described. The above description is illustrative and not limiting. Accordingly, it will be apparent to those skilled in the art that modifications may be made to the invention described above without departing from the scope of the following claims. For example, features of one embodiment may be combined with features of another embodiment. Other aspects of the present invention are described in the following numbered clauses:

[0087] 1. A method of controlling an actuator, comprising: providing a sequence of measurement samples (x(t)) at a first frequency; processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); downsampling at least the sequence of measurement samples to a second frequency lower than the first frequency; transmitting a sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples; using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency; A method comprising: 2. A method for controlling a real-time application, comprising: providing a sequence of measurement samples (x(t)) at a first frequency; processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); downsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a second frequency lower than the first frequency; using the sequence of down-sampled measurement samples and at least one sequence of down-sampled derivative measurement samples to provide a sequence of control error samples and at least one sequence of derivative control error samples; providing at least a sequence of control error samples using the sequence of control error samples and at least one sequence of derivative control error samples; using the sequence of control error samples to provide a sequence of real time application control samples at a third frequency greater than the second frequency; A method comprising: 3. The method of clause 1 or 2, wherein the step of processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples occurs substantially at the first frequency. 4. The method of any one of clauses 1 to 3, including processing the sequence of measurement samples at a first frequency to provide at least two sequences of derivative measurement samples. 5. Providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples, using the sequence of measurement samples to determine a sequence of control error samples at a second frequency; using the at least one sequence of derivative measurement samples to determine at least one sequence of derivative control error samples at a second frequency; and upsampling and filtering the sequence of control error samples and at least one of the sequence of derivative control error samples to a third frequency to provide a sequence of actuator or real-time application control samples; 3. The method of any one of clauses 1 to 4, including: 6. The method of any one of clauses 1 to 5, wherein the step of using the sequence of derivative measurement samples to determine at least one sequence of derivative control error samples includes the step of comparing the at least one sequence of derivative measurement samples with at least one sequence of derivative reference samples. 7. Providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples, comprising: upsampling and filtering the sequence of measurement samples and at least one sequence of derivative measurement samples to a third frequency greater than the second frequency to provide at least a sequence of upsampled measurement samples; using the sequence of upsampled measurement samples to determine a sequence of control error samples; 3. The method of any one of clauses 1 to 4, including: 8. The method of any one of clauses 1 to 7, wherein the step of using the sequence of measurement samples to determine the sequence of control error samples comprises the step of comparing the sequence of measurement samples with a sequence of reference samples. 9. The method of any one of clauses 1 to 8, wherein the third frequency is in the range of 100 kHz to 5 MHz. 10. The method of any one of clauses 1 to 9, wherein the first frequency and the third frequency are of the same order of magnitude. 11. The method of clause 10, wherein the second frequency is an order of magnitude smaller than the first frequency and / or the third frequency. 12. The method of any one of clauses 1 to 11, wherein the step of using a downsampled sequence of either a) at least one sequence of measurement samples and derivative measurement samples, or b) a combination of at least one sequence of control error samples and derivative control error samples to provide at least a sequence of control error samples comprises the steps of upsampling and using the sequence of upsampled samples in an nth-order hold filtering step, where n is an integer greater than or equal to 1. 13. A system for controlling an actuator, comprising: a measurement section for providing a sequence of measurement samples (x(t)) at a first frequency and for processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); a downsampler for downsampling at least the sequence of measurement samples to a second frequency lower than the first frequency; a control section for transmitting a sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; processing means for using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples to provide at least a sequence of control error samples, and for using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency; A system equipped with 14. A system for controlling a real-time application, comprising: a measurement section for providing a sequence of measurement samples (x(t)) at a first frequency and for processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); a downsampler for downsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a second frequency lower than the first frequency; a control section for providing a sequence of control error samples and at least one sequence of derivative control error samples using the sequence of downsampled measurement samples and the at least one sequence of downsampled derivative measurement samples; processing means for using the sequence of control error samples and at least one sequence of derivative control error samples to provide at least a sequence of control error samples, and for using the sequence of control error samples to provide a sequence of real time application control samples at a third frequency greater than the second frequency; A system equipped with 15. A system according to clause 13 or 14, wherein the processing means comprises a subtractor for comparing the input with at least one sequence of reference samples to provide a sequence of control error samples. 16. The processing means: an upsampler for upsampling at least one input sequence and providing at least one upsampled output sequence; a filter for filtering the at least one upsampled output sequence to provide a sequence of actuator or real-time application control samples; 16. The system of any one of clauses 13 to 15, comprising: 17. A position control system provided with at least one system according to any one of clauses 13 to 16. 18. An exposure apparatus, such as a lithography apparatus, provided with at least one system according to any one of clauses 13 to 16.

Claims

1. 1. A method of controlling an actuator, comprising: providing a sequence of measurement samples (x(t)) at a first frequency; processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); downsampling at least the sequence of measurement samples to a second frequency lower than the first frequency; transmitting the sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples to provide at least a sequence of control error samples; using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency; A method comprising:

2. 1. A method for controlling a real-time application, comprising: providing a sequence of measurement samples (x(t)) at a first frequency; processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); downsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a second frequency lower than the first frequency; using the sequence of downsampled measurement samples and the at least one sequence of downsampled derivative measurement samples to provide a sequence of control error samples and at least one sequence of derivative control error samples; using the sequence of control error samples and at least one sequence of derivative control error samples to provide at least a sequence of control error samples; using the sequence of control error samples to provide a sequence of real-time application control samples at a third frequency greater than the second frequency; A method comprising:

3. 3. The method of claim 1 or 2, wherein processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples occurs substantially at the first frequency.

4. 4. The method of claim 1, further comprising processing the sequence of measurement samples at the first frequency to provide at least two sequences of derivative measurement samples.

5. providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples, using the sequence of measurement samples to determine the sequence of control error samples at the second frequency; using the at least one sequence of derivative measurement samples to determine at least one sequence of derivative control error samples at the second frequency; and upsampling and filtering the sequence of control error samples and at least one of the sequence of derivative control error samples to the third frequency to provide the sequence of actuator or real-time application control samples; 5. The method of claim 1, comprising:

6. 6. The method of claim 1, wherein using the sequence of derivative measurement samples to determine at least one sequence of derivative control error samples comprises comparing the at least one sequence of derivative measurement samples with at least one sequence of derivative reference samples.

7. providing at least a sequence of control error samples using the sequence of downsampled measurement samples and at least one sequence of derivative measurement samples, upsampling and filtering the sequence of measurement samples and at least one sequence of derivative measurement samples to a third frequency greater than the second frequency to provide at least a sequence of upsampled measurement samples; using the sequence of upsampled measurement samples to determine the sequence of control error samples; 5. The method of claim 1, comprising:

8. 8. The method of claim 1, wherein using the sequence of measurement samples to determine a sequence of control error samples comprises comparing the sequence of measurement samples with a sequence of reference samples.

9. 9. The method of claim 1, wherein the third frequency is in the range of 100 kHz to 5 MHz.

10. 10. The method of claim 1, wherein the first frequency and the third frequency are of the same order of magnitude.

11. The method of claim 10 , wherein the second frequency is an order of magnitude smaller than the first frequency and / or the third frequency.

12. 12. The method of claim 1, wherein the step of providing at least a sequence of control error samples using a downsampled sequence of either a) a combination of at least one sequence of measurement samples and the derivative measurement samples, or b) a combination of at least one sequence of control error samples and the derivative control error samples comprises the steps of upsampling and using the sequence of upsampled samples in an n-th order hold filtering step, where n is an integer greater than or equal to 1.

13. 1. A system for controlling an actuator, comprising: a measurement section for providing a sequence of measurement samples (x(t)) at a first frequency and for processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); a downsampler for downsampling at least the sequence of measurement samples to a second frequency lower than the first frequency; a control section for transmitting the sequence of downsampled measurement samples and at least one sequence of downsampled derivative measurement samples; processing means for using the sequence of down-sampled measurement samples and at least one of the sequences of derivative measurement samples to provide at least a sequence of control error samples, and for using the sequence of control error samples to provide a sequence of actuator control samples at a third frequency greater than the second frequency.

14. 1. A system for controlling real-time applications, comprising: a measurement section for providing a sequence of measurement samples (x(t)) at a first frequency and for processing the sequence of measurement samples to provide at least one sequence of derivative measurement samples (x'(t), x''(t)); a downsampler for downsampling the sequence of measurement samples and at least one sequence of derivative measurement samples to a second frequency lower than the first frequency; a control section for using the sequence of downsampled measurement samples and the at least one sequence of downsampled derivative measurement samples to provide a sequence of control error samples and at least one sequence of derivative control error samples; processing means for using the sequence of control error samples and at least one sequence of derivative control error samples to provide at least a sequence of control error samples, and for using the sequence of control error samples to provide a sequence of real time application control samples at a third frequency greater than the second frequency; A system equipped with

15. 15. A system according to claim 13 or 14, wherein the processing means comprises a subtractor for comparing an input with at least one sequence of reference samples to provide the sequence of control error samples.

16. The processing means an upsampler for upsampling at least one input sequence and providing at least one upsampled output sequence; a filter for filtering the at least one upsampled output sequence to provide a sequence of actuator or real-time application control samples; 16. The system of any one of claims 13 to 15, comprising:

17. A position control system provided with at least one system according to any one of claims 13 to 16.

18. 17. An exposure apparatus, such as a lithographic apparatus, provided with at least one system according to any one of claims 13 to 16.

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