Richness and excitation on an interval-with application to continuous-time adaptive control

Richness and excitation on an interval is presented as a useful tool, which offers flexibility and mathematical simplification of the excitation issue. Feedback laws are considered which, when they are applied to any controllable time-invariant linear system, result in a closed-loop system such that the feedback signal, the measure of excitation which it produces, and the state vector of the system are all of the same order of magnitude on an interval. Such feedback laws generate excitation of time-varying linear systems, provided these systems are almost time-invariant and controllable in the interval of interest. Using these results, a smooth adaptive controller which controls and adapts simultaneously all the time and stabilizes and asymptotically regulates an arbitrary time-invariant controllable linear system of known order is obtained