Stochastic parallel model adaptation: theory and applications to active noise canceling, feedforward control, IIR filtering, and identification

General stochastic parallel model adaptation problems that consist of an unknown linear time-invariant system and a partially or wholly tunable system connected in parallel, with a common input, are considered. The goal of adaptation is to tune the partially tunable system so that its output matches that of the unknown system, despite the presence of any disturbance which is stochastically uncorrelated with the input. The general formulation allows applications to adaptive feedforward control and adaptive active noise canceling with input contamination, in addition to output error identification and adaptive IIR filtering. It is shown that in all the applications, the goal of adaptation is met whenever a matching condition and a positive real condition are satisfied. A special case of the results therefore resolves the long-standing problem of the convergence and the unbiasedness of the output error identification scheme in the presence of colored noise. A simple general technique for analyzing the strong consistency of parameter estimation with projection is also developed