Convergence and rate of convergence of a recursive identification and adaptive control method which uses truncated estimators

A stochastic approximation-like method is used for the recursive identification of the coefficients in
y_{n}=\sum\min{1}\max{l_{1}}a_{i}y_{n-i}+\sum\min{0}\max{l_{2}}b_{i}u_{n-i}+ \sum\min{1}\max{l_{3}}c_{i}w_{n-i}+w_{n}
, where
{w_{n}}
is a sequence of mutually independent and bounded random variables, and is independent of the bounded
{u_{n}}
. Such methods normally require the recursive estimation of the "residuals" or the
{w_{n}}
, and the algorithms for doing this can be unstable if the parameter estimates are not close enough to their true values. The problem is solved here by use of a simple truncated estimator, which is probably what would be used in implementation in any, case. Then, under a stability, and strict positive real type condition, with probability 1 (w.p.1) convergence is proved and the rate of convergence is obtained. An associated adaptive control problem is also treated.