Convergence of adaptive control and prediction algorithms with and without truncation on their state estimates and controls

In this paper, we prove the almost sure convergence of the proposed adaptive control and prediction algorithms. The algorithms involve a weighted least squares parameter estimator and the truncation on the state estimates and controls. For the algorithm with no truncation, we prove that the tracking error in controlling a plant with N-units of delay, converges to its optimum value in a certain average sense under the usual minimum phase and passivity assumptions of the literature. With an additional regularity assumption expressed only in terms of the desired trajectory and the "white" noise term, it is also proved that the tracking error also converges in a strong sense at an asymptotically arithmetic rate. Under this regularity condition it is also demonstrated that after almost a finite number of iterations the algorithm affords equal weightings to all the measurements. Under an additional persistency condition it is also shown that the parameter estimation error converges to zero at a rate specified by the degree of excitation. When the above algorithm involves truncation on the state estimates and controls, it is also proved that the required passivity condition is considerably weakened.