Interlacing Is Not Necessary for Adaptive Control

The stability and optimality of an adaptive minimum variance controller for stochastic systems was first established by Goodwin, Ramadge and Caines [1] for unit delay systems. Subsequently this was generalized by Goodwin, Sin and Saluja [2] to the general delay case, by using a direct approach and an interlaced multiple recursion. However, the interlaced algorithm is both practically cumbersome and theoretically unsatisfying. Hence it has been a long standing problem to determine whether interlacing is really necessary. We resolve this issue by showing that interlacing is not needed for the stability and optimality of direct adaptive minimum variance control based on either the stochastic gradient or the extended least squares algorithms. The results are immediately applicable to adaptive prediction.