In this paper, we examine alternative methods for reducing the dimensionality of nonlinear dynamical system models arising in control of rapid thermal chemical vapor deposition (RTCVD) for semiconductor manufacturing. We focus on model reduction for the ordinary differential equation model describing heat transfer to, from, and within a semiconductor wafer in the RTCVD chamber. Two model reduct...

This dissertation is devoted to the development and study of new techniques in the field of model reduction for large-scale linear time-invariant (LTI) dynamical systems. The behavior of processes in electrical networks, mechanics, aeronautics, civil engineering, micro-electro-mechanical-systems, weather prediction and many others can be described by mathematical models. Such models are usually...

(perhaps omitting quantum mechanics). The significance of this observation for hybrid systems is that the nominal description involving discontinuity is merely a convenient approximation at the relevant time scale which involves the (unmodelled?) neglect of dynamics on any faster time scales — whose details are then necessarily lost in the process of model reduction. We will argue that some res...

Roughly speaking, an “optimization problem” is the task of finding an element x = xo of a given set X for which the value Φ(x) of a given function Φ : X 7→ R is minimal. Alternatively, the objective could be to find an element xγ ∈ X such that Φ(xγh) < γ, where γ ∈ R is a given threshold, or to give evidence to non-existence of such xγ does not exist. Among the most familiar optimization proble...

Flexible manipulators can be characterized by a dynamic model with a large number of vibration modes, and the use of the model in the model-based control schemes requires reduction of model order. Balanced truncation is an effective method for model reduction of asymptotically stable systems by transforming the states to a coordinate system in which the controllability and observability Gramian...

We discuss linear time-invariant systems (LTI systems) of type ẋ = Ax + Bu, y = Cx + Du, where u is an input function, x is the state variable and y is the output of the dynamical system. Often one is only interested in the input-output properties of this system, which are of small size, whereas the state variable is typically large. Model order reduction (MOR) methods deal with these kind of s...

Contents 1. Glossary Range space: A column vector x such that x = Ay for some column vector y is said to be in the range space of matrix A, i.e., x ∈ R (A). Null space: A column vector x such that Ax = 0 is said to be in the right null space of matrix A, i.e., y ∈ N R (A). A row vector x such that xA = 0 is said to be in the left null space of A, i.e., y ∈ N L (A). In this text the symbol (⊥) i...

The finite element method (FEM) with model reduction is widely used to simulate deformable objects. Improving the performance and rendering quality of deformable objects simulated using FEM with model reduction has been a critical problem for several years. As computer technologies rapidly evolve, GPUs have seen significant improvement. It is clear that modern graphics processing units (GPUs) p...

Understanding the collective behavior of complex systems from their basic components is a difficult yet fundamental problem in science. Existing model reduction techniques are either applicable under limited circumstances or produce "black boxes" disconnected from the microscopic physics. We propose a new approach by translating the model reduction problem for an arbitrary statistical model int...