The strong consistency of the stochastic gradient algorithm of adaptive control

By use of a technique due to Chen, sufficient conditions are established for the strong consistency of the stochastic gradient (SG) algorithm for MIMO ARMAX stochastic systems without monitoring. This result is then used in conjunction with the method of disturbed adaptive controls introduced in [1], [2]. Hence it is shown that the SG algorithm generates strongly consistent parameter estimates while it is operating as a part of the SG algorithm of adaptive control of Goodwin, Ramadge, and Caines.