Almost sure convergence of adaptive identification prediction and control algorithms

The paper is concerned with the almost sure convergence of the adaptive parameter estimation, N-step ahead prediction and control algorithms based upon standard least square algorithm. With the usual stability and passivity assumptions for the prediction problem, it is demonstrated that the state estimation and the N-step ahead prediction errors converge to the optimum such errors achievable with known plant parameters, in the Cesaro sense. An additional regularity assumption on the signal model establishes the result that the state estimation and prediction errors also converge in the strong sense at an asymptotically arithmetic rate. Under an additional persistency of excitation condition it is shown that the parameter estimation error converges to zero at a rate specified by the degree of excitation. With the regularity condition holding, the convergence is also established for the adaptive control algorithms.