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.

An important feature of the facility we describe is that it uses severaltypes of sensing modalities including position sensing, tactile sensingand more conventional vision sensing. It can interact with objectsof different complexity and is subject to communication constraints arising in a completelynatural and generic way. In constructing it we have used off-the-shelfcomponents wherever possible and made choices with an eye towardflexibility and reliability.

In this research, an ULVAC ERA-1000 selective tungsten chemical vapor deposition system located at the University of Maryland was studied where a temperature difference as large as 120 ^oC between the system wafer temperature reading and the thermocoupled instrumented wafer measurement was found during the manual processing mode.

The goal of this research was to develop a simplified, but accurate, three-dimensional transport model that is capable of describing the observed reactor behavior.

A hybrid approach combining experimental and simulation studies was used for model development. Several sets of experiments were conducted to investigate the effects of process parameters on wafer temperature.

A three-dimensional gas flow and temperature model was developed and used to compute the energy transferred across the gas/wafer interface. System dependent heat transfer parameters were formulated as a nonlinear parameter estimation problem and identified using experimental measurements.

Good agreement was found between the steady-state wafer temperature predictions and experimental data at various gas compositions, and the wafer temperature dynamics were successfully predicted using a temperature model considering the energy exchanges between the thermocouple, wafer, and showerhead.

To increase the utility of existing simulators, model reduction methods must be used to extract the dominant spatial characteristics of the discharge; numerically efficient spectral projection methods are then used to generated the reduced model. These practical needs motivated the development of a set of simulation tools that provide a framework for process simulation, model reduction, and analysis of simulator predictions.

The goals of this thesis were to build this framework by identifying the computationally common elements of semiconductor device manufacturing process simulation, model reduction, and analysis methods, and to test these tools on the difficult problem of RF plasma simulation. The simulation tools were developed as a library of MATLAB functions; the library and demonstration scripts have been distributed through the MWRtools project website.

Our approach to this problem is based on plant diagonalization.To do this, we search for basis transformations for the vector of outputs coming from the sensors and the vector of inputs applied to the actuators so that, in the new bases, the MIMOsystem becomes a collection of decoupled SISO systems.This formulation provides a number of advantages for the synthesis and implementation of a feedback control law,particularly for systems where the number of inputs and outputs is large.

Of course, in order for this idea to be feasible,the required basis transformations must have properties which allow them to be implemented on a distributed control network. Namely, they must be computed in a distributed manner which respects the spatial distribution of the data(to reduce communication overhead) and takes advantage of the massive parallel processing capability of the network (to reduce computation time).

In this thesis, we present some tools which can be used to find suitable transforms which achieve "approximate"plant diagonalization. We begin by showing how to search the large collection of orthogonal transforms which are contained in the wavelet packet to find the one which most nearly, or approximately, diagonalizes a given real valued matrix.Wavelet packet transforms admit a natural distributed implementation,making them suitable for use on a control network.We then introduce a class of linear operators called recursive orthogonal transforms (ROTs) which we have developed specifically for the purpose of signal processing on distributed control networks.

We show how to use ROTs to approximately diagonalize fixed real and complex matricesas well as transfer function matrices which exhibit a spatial invariance property. Numerical examples of allproposed diagonalization methods are presented and discussed.

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.