Averaging with respect to arbitrary closed sets: closeness of solutions for systems with disturbances

We consider two different definitions of "average" for systems with disturbances: the "strong" and "weak" averages that were introduced by Nesic et al. (1999). Our definitions are more general than those given by Nesic et al., as we use the distance to an arbitrary closed set /spl Ascr/ instead of the Euclidean norm for states in the definitions of averages. This generalization allows one to deal with more general cases of averaging for systems with disturbances, such as partial averaging. Under appropriate conditions, the solutions of a time-varying system with disturbances are shown to converge uniformly on compact time intervals to the solutions of the system's average as the rate of change of time increases to infinity.