Some Function-Theoretic Issues in Feedback Stabilisation

This article aims to present for a mathematical audience some interesting function theory elaborated over recent years by control engineers in connection with the problem of robustly stabilising an imperfectly modelled physical device. The theory of feedback extends over a broad field, from rarefied differential geometry to down-to-earth nuts-and-bolts engineering. Function theory enters into the study of a class of problems of great practical importance, those relating to linear time-invariant systems. It is still the case that the great majority of engineering devices are modelled by such systems, and function theory remains an important strand in recent engineering researches on the stabilisation of uncertain systems. The connection with H and Hankel operators is widely known by now, but the extent to which engineers have developed the mathematical theory along novel lines deserves publicity. Challenges of an engineering nature have given rise to some beautiful ideas and results in function theory, and the purpose of this expository article is to present some notions arising from studies of robust stabilisation which deserve the attention of mathematicians. These notions relate to certain spaces of functions on the real line or subsets of the complex plane and to sundry metrics on these spaces which measure closeness of functions from the point of view of stabilisability. Many engineers have contributed to these developments, notably M. Vidyasagar, T. T. Georgiou, M. C. Smith, K. Glover, D. C. McFarlane, and G. Vinnicombe. Virtually everything in this paper is from [Vidyasagar 1984; Georgiou and Smith 1990; 1993; McFarlane and Glover 1990; Vinnicombe 1993; Curtain and Zwart 1995]. A useful textbook covering the elements of H-control is [Doyle et al. 1992]. However, these sources assume 1991 Mathematics Subject Classification. Primary 93D21, Secondary 30E10.