This paper considers $A_\infty$-algebras whose higher products satisfy an analytic bound with respect to a fixed norm. We define a notion of right Calabi--Yau structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich--Soibelman, and yields a flexible method for obtaining analytic potentials of Hua-Keller. We apply these results to the endomorphism DGAs of polystable sheaves considered by Toda, for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms on a projective variety. As a result, we can define a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This shows that the results of Toda can be extended to the quasi-projective setting, and yields a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi--Yau varieties.