During the period 1980-1983, I became interested in analytic interpolation and its relevance in engineering disciplines such as circuit theory, stochastic processes and optimization. It was the theory of the moment problem and orthogonal polynomials that first came to my attention, and which I explored in my Ph.D. thesis on ``partial realization of covariance sequences''. This is a topic of key importance in spectral estimation as well as in model identification. It was to this same topic that I focused attention again in recent years (1998-present), much of it in collaborative reseach with Chris Byrnes (Washington University) and Anders Lindquist (KTH, Sweden). This recent work has led to new ways of estimating power spectra based on finite (and typically short) observation records. The key feature of these new techniques is the ability to tune in on a particular harmonic interval and thereby allow for resolution significantly higher than pre-existing state-of-the-art. They may prove of importance in speech coding, radar/sonar signal processing, and medical imaging where we are currently focusing our research efforts.
Right after completing my Ph.D., I became interested in the subject of feedback theory. One of the key questions that was raised by the late George Zames had to do with describing and quantifying the type of modeling uncertainty that can be tolerated in a feedback system. In other words, one would like to have a quantitative notion of distance between dynamical systems (i.e., a metric) along with a computable robustness margin such that, a feedback system remains operational when the actual components deviate from the assumed models in the metric, by an amount less than the robustness margin. This led to a research program which advanced the concept of the gap metric as tool in the theory of robust control. My research on this topic was carried in collaboration with Malcolm C. Smith (University of Cambridge). Over the years we developed theoretical tools and algorithms for computation of the gap metric between systems, optimal controller design, approximation, identification, and control of flexible structures and distributed systems, where in each instance uncertainty, errors, and margins, refer to quantities measured using the gap metric. We carried out application studies on experimental facilities at JPL and at Wright Patterson Air Force Base. We have extended the concept of such a metric to nonlinear systems (since the early work addressed primarily linear systems). Work in progress focuses on systems with periodic oscillatory behavior. The importance of the gap rests on the fact that it gives a "representation-independent" notion of distance between systems which is natural and useful in the context of feedback theory.
Over the years I have also worked on control of periodic and sampled-data systems, on numerical problems of spectral factorization, quadratic optimal control, filtering, and finally, on modeling the cardiovascular system based on EKG and blood pressure data (an interesting example of a natural oscillatory system which is supposed to work continuously for 80+ years -- hopefully with no maintenance and certainly with no down-time).
Recent reports, computer code, seminars, and miscellaneous.