Modeling and Control of Distributed Thermal Systems

This paper investigates the application of model-based control design techniques to distributed temperature control systems. Multivariable controllers are an essential part of modern-day rapid thermal processing (RTP) systems. We consider all aspects of the control problem beginning with a physics-based model and concluding with implementation of a real-time embedded controller. The thermal system used as an example throughout is an RTP chamber widely used in semiconductor wafer processing. With its exceptionally stringent performance requirements (low nonuniformity of wafer temperature, high temperature ramp rates), RTP temperature control is a challenging distributed temperature control problem. Additionally, it is important in the semiconductor industry because of the progressively smaller “thermal budget” resulting from ever decreasing integrated circuit dimensions. Despite our emphasis on faster cold wall single-wafer processing RTP chambers, the approach described here for solving distributed temperature control problems is equally applicable to slower distributed thermal systems, such as hot-wall batch-processing furnaces. For the physical model, finite volume techniques are used to develop high-fidelity heat transfer models that may be used for both control design and optimal chamber design. Model-order reduction techniques are employed to reduce these models to lower orders for control system design. In particular, principal orthogonal decomposition techniques have been used to derive low order models. Techniques such as linear quadratic Gaussian H₂/H_∞ methods are employed for feedback control design. While the methods are illustrated here using a generic RTP system, they have been successfully implemented on commercial RTP chambers.