Multivariable feedback relevant system identification of a wafer stepper system

This paper discusses the approximate and feedback relevant parametric identi cation of a positioning mechanism present in a wafer stepper. The positioning mechanism in a wafer stepper is used in chip manufacturing processes for accurate positioning of the silicon wafer on which the chips are to be produced. The accurate positioning requires a robust and high performance feedback controller that enables a fast through put of silicon wafers. A set of multivariable nite dimensional linear time invariant discrete time models will be estimated, that is suitable for model-based robust control design of the positioning mechanism. INTRODUCTION Wafer steppers combine a high accuracy positioning and a sophisticated lithographic process to manufacture integrated circuits (chips) via a fully automated process. By means of a photolithographic process, the chip architecture is exposed on the surface of a wafer, a silicon disk covered with photo resist. In the application discussed in this paper, the wafer is supposed to carry approximately 80 chips. In order to expose the surface of the wafer, each chip is processed sequentially. Such a sequential process is needed as only one mask of the chip layout is available during the exposure fase of the photolithographic process. For that purpose, the wafer is placed on a moving table that needs to be moved (stepped) in 3 Degrees Of Freedom (3DOF) accurately for the sequential processing of the chips on the wafer. The work of Raymond de Callafon is nancially supported by the Dutch Systems and Control Theory Network. Author to whom all correspondence must be addressed Clearly, both the accuracy and the speed of the servo mechanism during the subsequent steps of the wafer will inuence the success and throughput of the production process of the chips on the wafer. Sophisticated control of this (multivariable) servo mechanism can help in achieving a required throughput by designing a multivariable feedback controller that is able to satisfy high performance requirements (de Roover et al., 1996). A model that describes the dynamical behaviour of the servo mechanism is needed to design such a controller thoughtfully. A dynamical model can be obtained by rst principle modelling, see e.g. de Roover and van Marrewijk (1995). Although such a model provides valuable knowledge of the dynamical behaviour, either the numerical completion of speci c elements in the servo system is undiscoverable or deliberate assumptions are posed to simplify the modelling. This causes the model to deviate from the actual dynamical behaviour of the system. Alternatively, a system identi cation procedure can be exploited in which experimental data is used directly. In this way, a model describing the dynamical behaviour is evaluated directly on the basis of the data coming from the actual system (Ljung, 1987). Although both modelling procedures provide insight in the dynamical behaviour of the positioning mechanism present in a wafer stepper, it is impossible to exactly characterize all phenomena describing the dynamics. On the one hand exact modelling can be impossible or too costly, on the other hand control design methods can get unmanageable if they are applied to models of high complexity. As a result, the model obtained is only an approximation of the system to be controlled. As the validity of any approximate 1 Copyright c 1997 by ASME model hinges on its intended use, the modelling procedure being applied should take into account the intended use of the model; control design. MODELLING FOR CONTROL In this paper the attention is focused on deriving Finite Dimensional Linear Time Invariant (FDLTI) models via system identi cation techniques that approximates the dynamical behaviour of the positioning mechanism in a wafer stepper. For an existing servo mechanism present in a wafer stepper, time domain observations are gathered to estimate models that can be used for subsequent controller design. The aim of this paper is to outline the system identi cation procedure being used and the performance improvement obtained when designing a multivariable controller. In order to estimate models suitable for control design, the following requirements should be satis ed. Preferably, the models should be a linear description of actual system to be controlled. In this way, standard tools for linear model-based control design can be used. Furthermore, control design methods become unmanageable if they are applied to models of high complexity. Hence, linear models should have a reasonable model order in order to formulate a manageable control design problem. As the models will be necessarily approximative, it should contain those dynamical aspects that are important for control design (Schrama, 1992b). Finally, the identi cation procedure being used should be able to deal with data that is obtained under closed-loop (controlled) conditions. This is due to the fact that many engineering systems are unable to operate without additional control, including the position servo mechanism of the wafer stepper. Estimating a linear model can be done by existing system identi cation techniques reported in the literature (Ljung, 1987; Soderstrom and Stoica, 1989) and available in the corresponding commercial software packages (Ljung, 1995). However, application of these techniques to nd models on the basis of closed-loop experiments that capture the dominant dynamical aspects relevant for feedback, is by far trivial. Estimating such models boils down to the fact that models, suitable for control design, can only be found by taking the closed loop operation of the model into account (Schrama, 1992a). In general, this leads to identication problem in which the criterion used for designing the subsequent controller should also be used to deduct the model. See for example the work by Zang et al. (1995) for LQG-based controller design. As the resulting model is just an approximation of the system to be identi ed, the controller based on the model has to be robust against any dissimilarities between the model and the system. This has been a motivation for the development of identi cation techniques that estimate an upper bound on the model error, see for example the contributions by Goodwin et al. (1992), Helmicki et al. (1993), Partington and Makil a (1995) Makila and Partington (1995) and the references therein. The resulting model error constitutes an allowable model perturbation around a nominal model being estimated and de nes a set of models where the actual system is assumed to be an element of. Subsequently, a robust controller can be designed on the basis of this set of models (Doyle et al., 1992). In this approach stability and performance requirements are guaranteed for the complete set of models, that includes the actual system to be controlled. The estimation of such a set of models for the design of a robust controller for the positioning mechanism of the wafer stepper is the main item in this paper. In order to estimate such a set of models by the estimation of a (low complexity) nominal model along with its allowable model perturbation, the identi cation procedure discussed in this paper uses the algebraic framework of stable fractional model representations, similarly as in de Callafon et al. (1994) or Van den Hof et al. (1995). The reasoning to use such a fractional model representations is due to the ability to deal with both stable, unstable or marginally unstable systems, such as the positioning mechanism discussed in this paper. As such, this approach enables one to nd a set of feedback relevant models by estimating stable factorizations of a nominal model along with a stable perturbation on the allowable model perturbations. Furthermore, the fractional approach can deal with observations obtained under closed-loop (controlled) conditions relatively easily. WAFER STEPPER SERVO MECHANISM Description of servo mechanism The servo mechanism discussed in this paper is an integral part of the Silicon Repeater 3rd generation (SIRE3) wafer stepper. The moving table, called the wafer chuck, that needs to position the wafer, is equipped with a air bearing and placed on a large suspended granite block to reduce the e ect of external vibrations. The position of the wafer chuck on the horizontal surface of the granite block is measured by means of laser interferometry. A schematic overview of this servo mechanism is depicted in Figure 1. Relative movements of the wafer chuck are measured by determining the phase shift of the laser beams re ected on the mirror block depicted in Figure 1. As the horizontal plane allows three degrees of freedom, three laser measurements uniquely determine the horizontal position of the 2 Copyright c 1997 by ASME j 3 r r r r r j j 1 2