An augmented UD identification (AUDI) algorithm for system identification is developed by rearranging the data vectors and augmenting the covariance matrix of Bierman's UD factorization algorithm. The structure of the augmented information (covariance) matrix is particularly easy to interpret and it is shown that the AUDI algorithm is a direct extension of the familiar recursive least squares (RLS) algorithm. The proposed algorithm permits simultaneous identification of the model parameters plus loss functions for all orders from 1 to n at each time step with approximately the same calculation effort as «th order RLS. This provides a basis for simultaneous model order and parameter identification so that problems due to over- and under-estimation of model can be avoided. Based on its least-squares properties, numerical robustness, theoretical basis and the fact that it simultaneously estimates multiple models, the proposed AUDI algorithm is recommended for use in place of RLS and Bierman's UD factorization algorithms.