This paper deals with the asymptotic hyperstability of switched time-varying dynamic systems involving switching actions among linear time-invariant parametrizations in the feed-forward loop for any feedback regulator controller potentially being also subject to switching through time while being within a class which satisfies a Popov-type integral inequality. Asymptotic hyperstability is proved to be achievable under very generic switching laws if at least one of the feed-forward parametrization possesses a strictly positive real transfer function, a minimum residence time interval is respected for each activation time interval of such a parametrization and a maximum allowable residence time interval is simultaneously maintained for all active parametrizations which are not positive real, if any. In the case where all the feed-forward parametrizations possess a common Lyapunov function, asymptotic hyperstability of the switched closed-loop system is achieved under arbitrary switching.