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A common issue with many system identification problems is that the true input to the system is unknown. This paper extends a previously presented indirect modelling framework that deals with identification of systems where the input is partially or fully unknown. In this framework, unknown inputs are eliminated by using additional measurements that directly or indirectly contain information about the unknown inputs. The resulting indirect predictor model is only dependent on known and measured signals and can be used to estimate the desired dynamics or properties. Since the input of the indirect model contains both known inputs and measurements that could all be correlated with the same disturbances as the output, estimation of the indirect model has similar challenges as a closed-loop estimation problem. In fact, due to the generality of the indirect modelling framework, it unifies a number of already existing system identification problems that are contained as special cases. For completeness, the paper is concluded with one method that can be used to estimate the indirect model as well as an experimental verification to show the applicability of the framework. [ABSTRACT FROM AUTHOR]