The aim of the paper is to investigate some concepts of convergence in the class of perfect BL-algebras. Similarity convergence was developed by G. Georgescu and A. Popescu in the case of the residuated lattices, while the convergence with a fixed regulator was studied by Cernák for lattice-ordered groups and MV-algebras and by the author for residuated lattices. In this paper we study the similarity convergence and the convergence with a fixed regulator for the perfect BL-algebras. The main result is the construction of Cauchy completion of a perfect BL-algebra.