A Popov-theory-based survey on digital control of infinite-dimensional systems with unboundedness

Control by computers has become an everyday reality for more than a decade. With the advent and proliferation of microcomputers, the role played by digital control design techniques has become increasingly more important. In many industrial settings, the designer has to control physical entities such as temperature and fluid flow, and appropriate modelling often leads to distributed-parameter systems— systems which are defined on Hilbert spaces. It has become necessary, therefore, to develop extensions of many of the 'classical' digital control strategies for finitedimensional systems to an infinite-dimensional setting. A generalization, to infinitedimensional systems with bounded operators, of the finite-dimensional results on the digital LQ optimal-control problem was achieved more than 20 years ago by Lee et al. [25]. Since then, various finite-dimensional control problems have been addressed and solved; as one would expect, various approaches to and theories about those control problems have been taken and constructed. We refer here to the H optimal control problem when considered in the deterministic case, the LQG optimal-control problem when considered in the stochastic sense, and the H suboptimal-control problems. In this paper, we consider the aforementioned problems in the infinite-dimensional setting, and our approach to control is a digital one. From the rather wide variety of control theories that have emerged up to this time, we shall focus our attention upon the so-called Popov theory. Let us begin by explaining why we have chosen this option, i.e. why a discrete Popov-theory approach to the digital control of DP systems is a valuable one. There are two main reasons.