Simultaneous adaptive control and identification via the weighted least-square algorithm

In this note, we prove the almost sure convergence of the proposed adaptive control and identification algorithms involving a weighted least-square parameter estimator. The analysis presented here strengthens the previous results achieved in this context. The proposed algorithm and the analysis achieve the convergence of the tracking error in a strong sense at an asymptotically arithmetic rate. The stronger result is obtained under a regularity and persistency condition introduced in this note, which also ensures that with probability one, the algorithm affords equal weighting to all the measurements after a finite number of iterations. Under these conditions it is also shown that the parameter estimation error also converges to zero. In the context of SISO ARMAX models it is proved that the regularity and persistency conditions are satisfied, under mild mixing conditions on the noise and no pole zero cancellation in the plant input-output transfer function, by a random perturbation of the reference trajectory.