One of the key issues in modelling for condition monitoring is how to accommodate the level of detail of the model description in order to suit the diagnostic requirements. The paper addresses a two-stage modelling concept, which tends to result in highly accurate grey models by combining prior knowledge and recorded data. A class of continuous-time models linear in parameters is treated. An idea related to the generation of candidate model structures for a class of distributed parameter systems is suggested. It is shown that the resulting models take minimal parameterisation for a given quality of fit. To calculate bias-free estimates of model parameters, an almost straightforward instrumental variables approach (IVA) combined with data pre-filtering is employed. It is shown that the estimates produced by TVA are largely invariant of the bandwidth of the pre-filters (state variable filters). This greatly reduces the overall modelling effort and makes the whole concept applicable even to less experienced users. The concept is used to design a diagnostic system for the heat transfer process in a tyre production plant. An excerpt of the results obtained in operating records is given.