In this paper the authors develop a recursive algorithm for identification of a possibly overparameterised state-space model for time-varying linear systems. Each parameter of the model is either known, lies in a bounded region, or is unknown. The cost to be minimised, subject to the parameter constraints, is formed from the output prediction error together with penalty terms. The algorithm is adaptive with respect to the penalty terms. Furthermore, the paper develops a state-space balancing method consistent with structure, if possible, and also considers the introduction of a time-varying forgetting factor.