A generalized projection estimation algorithm

Given a linear regression model in discrete-time containing a vector of $p$ constant uncertain parameters, this paper addresses the problem of designing an exponentially convergent parameter estimation algorithm, even when the regressor vector is not persistently exciting (not even in a finite time interval). On the basis of the definition of lack of persistency of excitation of order $q$ for the regressor vector, $0\le q\le p$ (which coincides with the classical definition of persistency of excitation when $q=0$), a generalized projection estimation algorithm is proposed which guarantees global exponential convergence of the parameter estimation error and allows for the on-line computation of the order $q$ of the lack of persistency of excitation. When the lack of persistency of excitation is of order zero, global exponential convergence to zero of the parameter estimation error is obtained, recovering a well-known result and the projection estimation algorithm as a special case.