Institute for Systems Research Technical Reports

First, we consider the problem of computing the controllability energyfunction without numerically solving the family of optimal control problems,or the associated Hamilton-Jacobi-Bellman equation, implied in its definition.Stochastically excited systems play a major role in our methodology. Wepresent a stochastic method for computing an estimate of the controllabilityfunction, and show that in certain situations the method provides an exactsolution. The procedure is tested on applications via Monte-Carlo experiments.

Then, we address the problem of numerically determining a Morse transformationfor a function with non-degenerate critical point at 0. We develop analgorithm for computing the desired nonlinear transformation and estimatingthe neighborhood on which the transformed controllability function isquadratic.

In the literature, examples of applied nonlinear balancing have been limited topseudo-balancing of 2-dimensional gradient systems and noting that in the caseof linear systems the energy functions approach reduces to the usual setting ofgramians. We apply our approach to numerically derive, for the first time,balanced representations of nonlinear state-space models. In particular, wepresent applications to a forced damped pendulum system and a forced dampeddouble pendulum system.

We offer a method for computing a workingapproximation to the controllability energy function, one of the mainobjects involved in the method. Moreover, we show that for a class ofsecond-order mechanical systems with dissipation, under certain conditionsrelated to the dissipation, an exact formula for the controllabilityfunction can be derived. We then present an algorithm for a numericalimplementation of the Morse-Palais lemma, which produces a local coordinatetransformation under which a real-valued function with a non-degeneratecritical point is quadratic on a neighborhood of the critical point.

Application of the algorithm to the controllabilty function plays a key rolein computing the balanced representation. We then apply our methods andalgorithms to derive balanced realizations for nonlinear state-space modelsof two example mechanical systems: a simple pendulum and a double pendulum.

The second part (Chapter 5) deals with modeling of rapid thermal chemicalvapor deposition (RTCVD) for growth of silicon thin films, viafirst-principles and empirical analysis. We develop detailedprocess-equipment models and study the factors that influence depositionuniformity, such as temperature, pressure, and precursor gas flow rates,through analysis of experimental and simulation results. We demonstratethat temperature uniformity does not guarantee deposition thicknessuniformity in a particular commercial RTCVD reactor of interest.

In thethird part (Chapter 6) we continue the modeling effort, specializing to acontrol system for RTCVD heat transfer. We then develop and apply ad-hocversions of prominent model reduction approaches to derive reduced modelsand perform a comparative study.

The overallobjective is to improve manufacturing effectiveness for epitaxial growth ofsilicon and silicon-germanium (Si-Ge) thin films on a silicon wafer. Growthtakes place in the ASM Epsilon-1 chemical vapor deposition (CVD) reactor, aproduction tool currently in use at ESSS. Phase II project results includedevelopment of a new comprehensive process-equipment model capable ofpredicting gas flow, heat transfer, species transport, and chemical mechanismsin the reactor under a variety of process conditions and equipment settings.

Applications of the model include prediction and control of deposition rate andthickness uniformity; studying sensitivity of deposition rate to processsettings such as temperature, pressure, and flow rates; and reducing the use ofconsumables via purge flow optimization. The implications of varioussimulation results are discussed in terms of how they can be used to reducecosts and improve product quality, e.g., thickness uniformity of thin films. We demonstrate that achieving deposition uniformity requires some degree oftemperature non-uniformity to compensate for the effects of other phenomenasuch as reactant depletion, gas heating and gas phase reactions, thermaldiffusion of species, and flow patterns.

In Part I of this paper we provide an exposition of some techniques that are useful in finding models of reduced complexity for dynamical systems involving flows. The material presented here is not new. The techniques we discussare based on classical theory such as the Karhunen-Loeve expansion and the method of Galerkin, and the more recent concept of "coherent structures." They have been heavily exploited in a wide range of areas in science and engineering.

The attempt here is to present this collectionof important methods and ideas together, at a high level of detail, in coherent form, and in the context of model reduction for simulation and control. In this manner we lead in to Part II which illustrates theirusefulness in model reduction by applying them to some elementary examples of distributed parameter systems which are related to processes found in semiconductor manufacturing.