Run-to-run control methods based on the DHOBE algorithm

Abstract

Since process models are typically not known exactly in real problems, it is important to estimate the process parameters before one applies the optimal control to a process. In this paper, the Dasgupta–Huang optimal bounding ellipsoid (DHOBE) algorithm is employed to estimate process parameters in semiconductor process run-to-run (RtR) control. At each iteration, the DHOBE algorithm returns an outer bounding ellipsoid of the likely process parameter set. If the vector center of the ellipsoid is taken as the estimate of the process parameter vector, then a model-reference controller results; if the vector within the ellipsoid that produces the worst expected cost is taken as the process parameter estimate, then a worst-case controller results. These two methods are compared with other RtR control schemes: the exponentially weighted moving average (EWMA) method and the optimizing adaptive quality controller (OAQC). Simulation results show that the performance of the model-reference RtR controller based on the DHOBE algorithm is comparable to or better than that of the other two RtR controllers in some specific examples of semiconductor processes.

Introduction

In industries such as semiconductor manufacturing, limitations and costs impose a need for adjusting process parameters on a run-to-run (RtR) basis. This need has originated a collection of techniques called RtR control (Baras & Patel, 1996; Sachs, Hu, & Ingolfsson, 1995; Boning, Moyne, & Smith, 1995; Castillo & Yeh, 1998; Hankinson, Vincent, Irani, & Khargonekar, 1997; Ingolfsson & Sachs, 1993; Ning, 1996; Deng, 1999). RtR control is a form of discrete-time process control in which the product recipe with respect to a particular equipment process is modified ex situ, i.e. between equipment “runs”, so as to compensate for the effects of process drifts, large shifts, and other disturbances to keep the outputs at the prescribed target values. A drift disturbance, which may be produced by the equipment aging, change of environment or other factors, causes a slow and constant change of the outputs of a process. Different from a drift disturbance, a shift disturbance (step disturbance) causes a large change of the outputs of a process in a few runs. It may be produced by the failure of a component, change of the operator or other reasons.

Generally, a RtR controller is designed in the following way. First, it computes an optimal control based on the initial process model. The initial process model can be derived from former off-line experiments such as using the response surface model (RSM) method. The controller does not modify the recipe during a run. At the end of the current run, the controller updates the process model based on the new measurements. Finally, at the beginning of the next run, it adjusts the recipe according to the updated process model. A typical block diagram of a RtR controller is illustrated in Fig. 1. The reason why it generates the new recipe from the post-process measurements on a run-to-run basis is, on one hand, lack of on-line sensors for the process. On the other hand, frequent changes of inputs to the process may increase the variability of the process's output (Sachs et al., 1995). Sometimes, a deadband is utilized in order to make the model changes less frequent.

Currently, there are three main RtR control schemes available:¹ The exponentially weighted moving average (EWMA) method, the least-squares estimation (LSE) method and the set-valued RtR controller.

1. The EWMA method (Sachs et al., 1995) is widely used in RtR control for its simplicity and efficiency to compensate for smooth drifts and other small disturbances. The EWMA method uses a linear (affine) model to approximate a process and only updates the offset term in the model. There are many modifications to the EWMA controller. For example, a deadband may be added to the EWMA controller to further reduce the variation of the process output (Sachs et al., 1995). Statistical methods can be combined with the EWMA controller to identify the existence of a shift and estimate the shift's magnitude. Then a rapid mode can be applied by adjusting the model parameters quickly (Sachs et al., 1995). The weight of the EWMA controller is an important factor that affects its performance. In (Smith & Boning, 1997), the authors use a neural network to adjust the weight of the EWMA controller. The neural network has to be trained off-line before it is deployed. A double exponential forecasting filter, which has two EWMA modules, is also developed to predict and remove drifts in a process (Butler & Stefani, 1994).

2. The LSE method. Typical examples are the optimizing adaptive quality controller (OAQC) (Castillo & Yeh, 1998) and the Kalman filter based approach (Palmer, Ren, & Spanos, 1996). The OAQC uses a second-order model to approximate a process. It may have better performance than the EWMA controller in controlling a non-linear process. All the coefficients of the model in the OAQC can be adjusted from run to run (Castillo & Yeh, 1998). The Kalman filter based approach uses a linear model to describe a process. Different from an EWMA controller, the Kalman filter based method can adaptively adjust both the slope and the intercept terms (Palmmer et al., 1996). Therefore, the LSE method based RtR controllers may have stronger tracking ability than the EWMA controller.

3. The set-valued RtR control method (Baras & Patel, 1996). Due to measurement errors and environment noises, it is difficult to find an exact process model. The location of the likely process model parameters for the next optimization run form a set. We could be quite certain that the parameter vector is somewhere in this set. Due to disturbances, the exact model parameters are unknown. An outer-bounding ellipsoid is usually used to approximate the set of likely parameter values; the ellipsoid is used for its simplicity. In (Baras & Patel, 1996), they apply the optimal volume ellipsoid (OVE) algorithm (Cheung, Yurkovich, & Passino, 1991) to find the bounding ellipsoid. In this paper, we are going to apply a more general ellipsoid algorithm, the Dasgupta–Huang optimal bounded ellipsoid (DHOBE) algorithm (Dasgupta & Huang, 1987) to approximate the likely model parameter set. The DHOBE algorithm updates the parameter estimates only when the new measurements contain new information. This reduces significantly the computational load for the set-valued RtR control method to estimate the process model parameters. The DHOBE algorithm was improved in (Rao & Huang, 1993) by introducing a rescue procedure. When the process undergoes abrupt shifts and modeling errors, the rescue procedure improves the performance of the algorithm and accordingly that of the controller. Therefore, a DHOBE-algorithm-based controller may work well for large step disturbances and model errors, which may be hard for other control methods to compensate for. Moreover, the DHOBE-algorithm-based controller can use a nth-order polynomial model to approximate a process and adjust all the coefficients adaptively. Hence, it may even have stronger tracking performance than the LSE-method-based controllers.

At each iteration, the DHOBE algorithm returns an outer bounding ellipsoid that contains the true parameter vector with high probability. If the vector center of the ellipsoid is taken as the estimate of the parameter vector, the explicit model update is implemented and leads to a model-reference method. If we search for the worst expected output that may be produced by a vector within the ellipsoid and then minimize the worst-case cost, a set-valued worst-case controller is obtained.

The DHOBE algorithm and the design of the corresponding controllers will be introduced in Section 2. The DHOBE-algorithm-based controllers will be compared with the EWMA controller and the OAQC in Section 3. Section 4 includes our conclusions from our experiments/simulations.