System-theoretic and algebraic aspects of the rings of stable and proper stable rational functions

A new approach in transfer-function methods for solving a variety of control-theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices. The rings of stable and proper stable rational functions are well known to be Euclidean domains. These rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained. This major difficulty in extending the idea of polynomial models of Fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form. A natural realization theory is thus developed for matrix fraction representations of transfer-function matrices over these rings. As an application of the new theory developed, linear matrix equations (QX+RY=T and QX+YR=T) over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field.     

Skew-prime polynomial matrices: The polynomial-model approach

We examine the concept of skew-primeness of polynomial matrices in terms of the associated polynomial model. It is shown that skew-primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism. This basic result is then applied to the special case of nonsingular polynomial matrices. We investigate the nonuniqueness of skew-complements of a skew-prime pair. It is shown that the space of equivalence classes of skew-complements is in bijective correspondence with a finite-dimensional linear space. Finally, the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly.     

Real-Time Measurement of Thin Film Thickness During Plasma Processing

An in situ single point two-color laser interferometer is used to monitor in real-time the thickness of thin transparent films during processing. The instantaneous change of film thickness is determined by comparing the measured laser reflection interference to that calculated by a model. The etch or deposition rates of the film are determined within 1–2 seconds. The film thickness is also determined in real-time from the phase difference of the reflected laser intensity between the two laser colors. Use of two-color laser interferometry improves the accuracy of the calculated etch or growth rates of the film considerably. Moreover, the two colors provide a clear distinction between film etching and deposition, which may often occur during the same process, and can not be determined by a single color interferometer. The uniformity of the film's etch or deposition rates across the substrate is monitored by an in situ full-wafer image interferometer. The combined use of these two sensors provide instantaneous information of the film thickness, etch or growth rates, as well as time averaged uniformity of the process rates. This diagnostic setup is very useful for process development and monitoring, which is also suitable for manufacturing environment, and can be used for real-time process control.