When considering several identical and independent systems subject to failures and imperfect maintenances, generalized virtual age models can be used to describe the corrective maintenance process. When the systems are not strictly identical, we can make use of covariates to take into account their heterogeneity. These are known data which can eventually be dynamic and which may influence more or less the working of the systems. Each covariate is associated to a coefficient which measures in a sense the influence of this covariate. In this paper we propose a mathematical framework in order to add time-dependent covariates to a virtual age model. It allows to simulate data with a given model and to estimate the parameters associated both to the model and to the covariates. The estimation is performed thanks to the maximization of the log-likelihood, assuming piece-wise constant covariates. We also give an insight into the quality of estimation.