Church, Kevin E. M.Smith, Robert J.2018-04-202018-04-202018-01-01http://dx.doi.org/10.1016/j.jmaa.2017.08.026http://hdl.handle.net/10012/13152The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.jmaa.2017.08.026 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/The time-scale tolerance for linear ordinary impulsive differential equations is introduced. How large the time-scale tolerance is directly reflects the degree to which the qualitative dynamics of the linear impulsive system can be affected by replacing the impulse effect with a continuous (as opposed to discontinuous, impulsive) perturbation, producing what is known as an impulse extension equation. Theoretical properties related to the existence of the time-scale tolerance are given for periodic systems, as are algorithms to compute them. Some methods are presented for general, aperiodic systems. Additionally, sufficient conditions for the convergence of solutions of impulse extension equations to the solutions of their associated impulsive differential equation are proven. Counterexamples are provided.enAttribution-NonCommercial-NoDerivatives 4.0 InternationalExponential regulatorImpulse extensionImpulsive differential equationsRobust stabilityStabilityTime-scale toleranceArticle