New Issues in the Mathematics of Control
The field of automatic control has emerged over the last century as an indispensable part of technology and an important element in the conceptualization of scientific ideas in a variety of fields. Because of its wide applicability, many of its mathematic
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1. Introduction The field of automatic control has emerged over the last century as an indispensable part of technology and an important element in the conceptualization of scientific ideas in a variety of fields. Because of its wide applicability, many of its mathematical aspects have been explored broadly and refined to a high degree. Other avenues are just now beginning to be explored. As we enter the 21st century it seems appropriate to assess the progress that has been made and discuss what may lie in store. In this paper we review the applicability of existing theory in areas such as robotics, physics and biology and discuss the kinds of advances that would help make it possible for the subject to reach its full potential as an aid to science and technology. For the non-specialists, it may be useful to have in mind a few 20th century problems to use as points of departure. We attach a key to each so as to make it easy to refer back to an item, as needed. There is not an all encompassing reference for the topics touched here but the general references Astrom and Wittenmark [1], Polderman and Willems [2] and Sontag [3] cover considerable ground. A. Regulator Problems: Consider a variable, or set of variables, associated with a dynamical system. They are to be maintained at some desired values in the face of changing circumstances. There exist a second set of parameters that can be adjusted so as to achieve the desired regulation. The effecting variables are usually called inputs and the affected variables called outputs. Specific examples include the regulation of the thrust of a jet engine by controlling the flow of fuel, the regulation of the oxygen content of the blood using the respiratory rate and the control of a beam of particles in a particle accelerator by modulating magnetic fields.
B. End Point Control Problems: There are inputs, outputs and trajectories, as above. In this case the shape of the trajectory is not of great concern but rather it is the end point that is of primary importance. Standard examples include rendezvous problems such as one has in space exploration, batch processing in chemical engineering in which reactants are introduced and the process controlled in such a way as to make a desired product with a specified purity, control of nuclear spins in nuclear magnetic resonance through the application of magnetic fields and radio frequency pulses and the control of an electron beam in B. Engquist et al. (eds.), Mathematics Unlimited — 2001 and Beyond © Springer-Verlag Berlin Heidelberg 2001
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such a way as to have it hit the inside of a television tube at a location specified by the signal received by the antenna.
C. Servomechanism Problems: There are inputs, outputs and trajectories, as above, and an associated dynamical system. In this case, however, it is desired to cause the outputs to follow a trajectory specified by the input. Examples include the control of a milling machine so that it will remove metal along the path specified by the blueprints, the control
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