Minimum-variance and self-tuning control

In Chapter 6 we have studied LQG control system design for state-space models. Since ARMAX models can be realized in state-space form, the results apply equally to ARMAX models. In either case, it is supposed that the parameters of the model are precisely known. On the other hand, we have presented in Chapter 4 techniques for identifying unknown systems from input/output data. Is it possible to combine these techniques and design controllers for ‘unknown’ systems involving some kind of on-line combination of identification and control? The general area to control system design for imperfectly known (and possible time-varying) systems is known as adaptive control and has been the subject of extensive study over many years. In this chapter we do not attempt any overall coverage of this area (which would require at least a whole book in itself) but restrict ourselves to discussing two key ideas — minimum-variance control and self-tuning regulators — which are closely related to the material of the preceding chapters. Both of these ideas are in their present form due to K.J. Astrom and co-workers (1970,1973) and have since burgeoned into a minor industry (quite literally, in that computer controllers incorporating these concepts are now commercially available). We also discuss the related ideas of pole-shifting regulators, which retain more links to classical control system design, and were introduced by Wellstead and co-workers (1979).