Studies in system representation and control

Abstract

Representation of an abstract system by a mathematical object such as a matrix or a differential equation is important because much of the analysis is done via representations. For discrete-time input-output systems, infinite-matrix kernel representation is the most general representation. We use the notion of finite-horizon systems to characterize all operators defined on $\ell\sb{p}$ spaces, $p\in\lbrack1, \infty$) that can be represented by an infinite-matrix kernel. We obtain a characterization of complete and closed behaviors in terms of finite-horizon systems. We give a new definition of stabilizability and obtain a characterization of stabilizability of systems on Hilbert spaces in terms of closable operators. A robustly stabilizing controller stabilizes every member of the model class to which the given plant belongs. We show that if the model class is an open set in $H\sb\infty$, there is a necessary and sufficient condition for robust stability. But if the model class is a closed set, the expected condition is only sufficient but not necessary. Same results hold for robust performance as well. Also, there is an equivalence between robust stability and robust performance problems if the model class is open, but none if it is closed. However, even with open model class, the expected conditions may be only sufficient but not necessary for uniform stability and performance. We present necessary and sufficient conditions for uniform stability in certain cases. There is sometimes a need to eliminate sinusoidal vibrations in the body of a mechanical system such as a cryogenic cooler. These vibrations are modeled as output disturbances. We assume that we know the frequencies of these sinusoids, but not their phases. There are infinitely many controllers resulting in notch filters that eliminate the sinusoids asymptotically. A weighted combination of the energies of the error signal and the control signal is a good criterion of optimality. We reduce this optimization problem to a weighted $H\sb2$-minimization problem. Due to some technical reasons, this problem is more difficult to solve than the regular Linear Quadratic Gaussian problem. We present a solution that involves solving two Riccati equations and a stable projection.