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Stability and performance of hybrid integrator-gain systems (HIGS) are mostly analyzed using linear matrix inequalities (LMIs) to construct continuous piecewise quadratic (CPQ) Lyapunov functions. To create additional methods for the analysis of HIGS and other conewise linear systems, a method based on linear programming (LP) for constructing continuous piecewise affine (CPA) Lyapunov functions is investigated. In this paper, it is shown how linear programming can be used to prove input-to-state stability and calculate upper bounds on the L1-gain and H1-norm. The numerical efficiency of this CPA (LP) method will be compared to that of the CPQ (LMI) method on a numerical example involving HIGS.

The operation time of an ideal reliable wafer scanner model is defined at the die level where the actual exposure process takes place as the time unit per die, or at the wafer substrate level as the time unit per wafer substrate. Therefore, the machine throughput is given as the reciprocal of the operation time. The involved motion profiles of a machine, namely the step-and-scan trajectories, function as the heartbeats that drive its multidisciplinary elements, which suggests that a multidisciplinary design optimization should be involved when such profiles are selected or designed. This is also true when considering the traverse motion profiles among rows and columns within the wafer substrate. The step-and-scan trajectories affect the machine throughput, performance, and die yield. The effects of tracking such profiles appear as structural vibration, tracking errors, and thermal loading at various machine elements such as the actuators, the reticle, the wafer, and the projection elements specifically when the exposure high-energy duration and frequency are not taken into consideration while designing the reference motion. From the dynamics perspective, having a reference motion with nonzero and bounded higher-order derivatives is recommended since it enhances the tracking performance of the machine, however, its ability to increase the operation time is usually overlooked. In an attempt to understand such effects, we present a case study that outlines the aforementioned aspects using three step-and-scan profiles of mainly $3^{rd}$ -order. Taking the dynamics of the driven stage into consideration through input shaping, both the step-and-scan and traverse motion profiles are analyzed. We provide analytical expressions that can be used to generate both types of motion profiles on the fly without additional optimization. A simulation example of a simplified wafer scanner machine shows the usefulness of the proposed framework. Note to Practitioners—Choosing the most suitable operating conditions of a lithography machine is challenging. These conditions affect machine productivity, profit margin, and maintenance. In this paper, we reveal the relation between the selection of operating conditions based on several decision variables- and the kinodynamic step-and-scan trajectory generation based on specific machine parameters and clients’ requirements. Being chart-based, the selection process of an operating point can be less practical at some points. However, using appropriate curve fitting tools, the information provided in the optimal operating charts can be put into suboptimal closed-form expressions that facilitate the selection process. Therefore, the designed trajectories parameters can be easily saved in lookup tables for ease of evaluation and future use. This helps in accommodating changes in the operation plans and flexible manufacturing systems. Also, starting with a given set of machine parameters, it is possible to calculate the optimal machine operating point when the input shaping technique is used, as illustrated in this paper.

Flexible dynamics in motion systems lead to inherent spatio-temporal system behavior. The aim of this paper is to develop an unified approach for the Identification of modal models of spatio-temporal overactuated systems. The approach exploits the modal modeling framework and the overactuated setting to enhance the estimation of the spatial system behavior. The proposed approach is applied in an experimental case study. The case study considers an experimental overactuated stage and illustrates the effectiveness of the proposed approach.

This paper considers the class of control systems containing so-called split-path nonlinear (SPAN) filters, which are designed to overcome some of the well-known fundamental limitations in linear time-invariant (LTI) control. In this work, we are interested in developing tools for the stability analysis of such systems using frequency-domain techniques. Hereto, we explicitly show the equivalence between a set of linear matrix inequalities (LMIs) with S-procedure terms, guaranteeing stability of the closed-loop (SPAN) system, and a frequency-domain condition. We also provide a systematic procedure for verifying the frequency-domain condition in a graphical manner. The results are illustrated through a nummerical case study.

It is well-known that the performance of linear time-invariant (LTI) feedback control is hampered by fundamental limitations. In this paper, it is shown that by using a so-called hybrid integrator-gain system (HIGS) in the controller, important fundamental LTI performance limitations can be overcome. In particular, in this paper, we show this for two well-known limitations, where overshoot in the step-response of the system has to be present for any stabilizing LTI controller. For each case, it is shown that by using HIGS-based control, one can avoid overshoot in the step-response of the system. Key design considerations for HIGS-based controllers as well as the stability of the resulting closed-loop interconnections are discussed.

An experimental demonstration is given of a nonlinear feedback controller applied to a short-stroke wafer stage system of an industrial wafer scanner. The controller design adopts a classical linear PID-based configuration in which both the integrator and low-pass filter are replaced with hybrid integrator-gain-based filter elements. The reduced phase lag associated with these filters gives rise to increased design flexibility, and potentially enables a substantial increase in low-frequency disturbance suppression. The nonlinear controller is designed by means of a quasi-linear loop-shaping approach, while stability of the full nonlinear closed-loop system is verified by solving a set of linear matrix inequalities. Performance of the controlled system is discussed on the basis of measurement results obtained from a wafer stage system.

In this paper a linear bandpass filter is compared to a hybrid integrator-gain based bandpass filter regarding its usefulness in active vibration isolation. Vibration isolation refers to a form of skyhook damping in which a velocity output signal from a system having structural dynamics is fed back to a controller, the latter having bandpass characteristics. At those frequencies where the controller passes the input signal (after gain multiplication) a force proportional to velocity is obtained that can be used to provide active damping to the system, i.e., damping of one (or more) of its structural modes. Outside this frequency band the controller input signal is attenuated, which is often desirable in view of incorrect sensor information at low frequencies and/or to avoid amplification of high-frequency noise. In this context, the use of a hybrid integrator-gain system will be studied regarding its possible phase advantages compared to linear integrators. These advantages stem from the control design itself, in the sense that by design the control output force signal and the input velocity error signal have equal sign. In the context of vibration control, an enhanced transient (closed-loop) response is obtained, i.e., less overshoot and reduced settling times, but at the cost of increased rise times.

In an attempt to surpass Bode’s gain-phase relationship, a hybrid integrator-gain system is studied that has significantly less phase lag in its describing function description when compared to a linear integrator. The hybrid integrator is designed to obtain improved low-frequency disturbance rejection properties under double-integrator (PI2D-like) control, but without the unwanted increase of overshoot otherwise resulting from adding an extra linear integrator. Closed-loop stability of the hybrid control design is guaranteed on the basis of a circle-criterion-like argument and checked through (measured) frequency response data. Closed-loop performance is obtained by data-driven optimization using gradients derived from a state-space description of the hybrid integrator, frequency response data from the linear part of the control system, and data obtained from machine-in-the-loop measurement.

Active vibration isolation is essential for industrial high-precision systems to suppress both floor vibrations and direct disturbance forces. This paper proposes a systematic H∞feedback and feedforward controller design to suppress both types of disturbances with minimum amplification of sensor noise. Regarding the latter, parametric disturbance and noise models are used in the design. The resulting H∞controller is shown to have disturbance rejection properties similar to a manually tuned controller but with less noise amplification and actuator drift.

This paper proposes a practical tuning guide for mixed feedback and feedforward control design of active soft-mounted vibration isolators. A model is derived based on measured frequency response function data obtained from an industrial vibration isolator. Using this model, feedback and feedforward controllers are designed and evaluated. The mixed-sensor feedback controller uses a combination of relative position feedback and absolute acceleration feedback. By mixing the sensor signals, amplification of sensor noise is minimized. It is shown that feedback control can greatly enhance the disturbance rejection properties of disturbances that act directly on the payload of the machine. In addition, a feedforward controller is designed that improves the rejection of indirect disturbances on the machine, i.e. disturbances that are transmitted via the vibration isolator.

A reset integral controller is discussed that induces improved low-frequency disturbance rejection properties under double integrator control without giving the unwanted increase of overshoot otherwise resulting from adding an extra linear integrator. To guarantee closed-loop stability, a (conditional) reset condition is used that restricts the input-output behavior of the dynamic reset element to a [0,α]-sector with αa positive (finite) gain. As a result, stability can be guaranteed on the basis of a circle criterion-like argument and checked through (measured) frequency response data. Both stability and performance of the control design will be discussed via measurement results obtained from a wafer stage system of an industrial wafer scanner.

This paper gives an exposition of examples of data-based control and optimization that found their way to successful application in the motion systems of industrial wafer scanners. The examples represent selective works brought together in an overview that highlights the possibilities, challenges, and open issues in data-based motion control of wafer scanners.

The implementation of lightweight high-performance motion systems in lithography and other applications imposes lower requirements on actuators, amplifiers, and cooling. However, the decreased stiffness of lightweight designs increases the effect of structural flexibilities especially when the point of interest is not at a fixed location. This is for example occurring when exposing a silicon wafer. The present work addresses the problem of compliance compensation in flexible structures, when the performance location is time-varying. The compliance function is derived using the frequency domain representation of the solution of the partial differential equation (PDE) describing the structure. The method is validated by simulation results.

In this paper, we consider a hybrid system consisting of a motion stage that exhibits, from a control point-of-view, position-dependent dynamics as a result of varying sensor configurations. A scheduled controller design is proposed that is intuitive for control engineers as all individual filters can be obtained using classical frequency-domain loop-shaping techniques thereby connecting to the industrial motion control practice. Moreover, data-based graphical conditions are provided in the frequency-domain under which closed-loop stability can be verified irrespective of how the switching between the controllers occurs in time. The effectiveness of the proposed scheduling technique, as a way to improve both transient and steady-state performance compared to the current state-of-the-art industrial control solution, is demonstrated by means of experiments on a high-precision industrial motion stage.

A novel feedforward control strategy is presented to isolate precision machinery from broadband floor vibrations. The control strategy aims at limiting the low-frequency controller gain, to prevent drift and actuator saturation, while still obtaining optimal vibration isolation performance at higher frequencies. This is achieved by limiting the low-frequency control action such that almost no phase shift is introduced at higher frequencies. To minimize model uncertainties, the feedforward controller is implemented as a self-tuning IIR filter that estimates the parameters online. Only a few parameters have to be estimated, which makes the algorithm computationally efficient. An additional feedback controller is designed to make the self-tuning algorithm more robust. The effects of feedforward and feedback control add up. The control strategy is successfully validated on an experimental setup of a vibration isolator.

The aim of this paper is to extend iterative feedback tuning (IFT), which is a data- based approach for controller tuning, with robustness constraints. Hereto a constrained IFT problem is formulated that is solved by introducing a penalty function. Essentially, the gradient estimates decompose into (a) the well-known IFT gradients and (b) the gradients with respect to this penalty function. Experimental results obtained from the motion control systems of an industrial wafer scanner confirm enhanced performance with guaranteed robustness properties.

Estimation and control of unmeasurable performance variables in complex large-scale systems is an important issue in systems and control. The preferred solution to this problem is to have a relatively low-order and accurate standard plant model which can be used for control purposes. For this purpose, a two-step procedure is proposed. The first step is to generate a reduced Finite Element (FE) model based on the selection of desired Degrees Of Freedom (DOFs), resulting in reduced-order mass, damping, and stiffness matrices. The second step is updating of the reduced-order FE model that is carried out to minimize the differences between the model and the measurements from the structure with the focus on input-output behavior. The presented approach helps to create sufficiently accurate reduced-order dynamic models which can be used for control purposes. The approach will be examined on a planar plate FE model.

This paper discusses a method for introducing frequency-domain robustness constraints in a data-driven control design. The method is based on adaptive weighting filters used in a time-domain cost function to be minimized. In an iterative way, violation of the frequency-domain robustness constraints is checked by model prediction and accounted for by adapting the weighting filters. The method is applied to the stage synchronization problem in wafer scanners, which involves synchronization of the outputs of two motion systems, and will be discussed in view of both simulation results and experimental results.