The design of electromagnetic invisibility cloaks is based on singular mappings prescribing zero or infinite values for material parameters on the inner surface of the cloak. Since this is only approximately feasible, an asymptotic analysis is necessary for a sound description of cloaks. We adopt a simple and effective approach for analyzing electromagnetic cloaks - instead of the originally proposed singular mapping, nonsingular mappings asymptotically approaching the ideal one are considered. Scattering and radiation from this type of imperfect cylindrical cloaks is solved analytically and the results are confirmed by full-wave finite element simulations. Our analysis sheds more light on the influence of this kind of imperfection on the cloaking performance and further explores the physics of cloaking devices.