The role of convexity in the adaptive control of rapidly time-varying systems

In classical adaptive control the parameters are assumed to be fixed or slowly time-varying. In order to facilitate parameter estimation/tuning it is desirable to have the set of admissible parameters lie in a convex set; if this set is not convex, a common trick is to replace it with its convex hull, but the adaptive control problem is challenging if stabilizability of the set of admissible parameters is lost. However, such a convexity assumption is an artifact of the approach to the problem, rather than an inherent constraint, since most logic-based and supervisory approaches to the problem make no such requirement. On the other hand, here we show that losing stabilizability on the convex hull of the set of admissible parameters plays an important role in the adaptive control of rapidly time-varying systems.