Control and optimization with dimensionality constraints

Abstract

The purpose of this thesis is to develop synthesis tools for control design with dimensionality constraints. In particular, given a model for a physical process, the goal is to characterize all possible controllers of a certain dimension which satisfy given performance criteria. In classical feedback design, the complexity of controller adversely affects robustness of the regulatory mechanisms of the feedback and adds to the fragility of the system. The complexity is often due to the difficulty in imposing performance specifications in a natural mathematical context. Typically, this is done using "weight functions" which encapsulate the specifications, and then introducing those in a suitable optimization problem. A contribution of this work is to address a certain type of optimization problem and the choice of weight functions. More precisely, we develop a new design approach which leads to a controller achieving both requirements, the performance specifications and low complexity, at the same time. Further, this thesis generalizes the previous methods for multivariable systems by developing analogous theory and techniques. The main contribution in multivariable analytic interpolation is to characterize a family of minimal McMillan degree solutions by a choice of spectral-zero dynamics. In addition to application of this theory for model-matching in control design, we show how to use the same techniques for maximum power transfer in circuit theory, and for spectral estimation in signal analysis. Also in this thesis we give new results on implementation of controllers with some very specific elements. One such, which is hard to simulate on a digital computer, is what could be described as "half capacitor". It implements a "fractional integration" and can be used to a great advantage in classical feedback design, providing gain but without introducing time-lag.