Exponential Resetting and Cyclic Resetting Recursive Least Squares

We present two extensions of recursive least squares (RLS) with exponential forgetting (EF), namely, exponential resetting (ER) RLS and cyclic resetting (CR) RLS. Both methods guarantee that the covariance matrix is bounded above and below in the absence of persistent excitation. Under zero excitation, ER-RLS guarantees convergence of the covariance matrix
$P_{k}$
to a user-designed positive-definite matrix
$P_{\infty }$
. However, ER-RLS is more computationally complex than EF-RLS. In contrast, CR-RLS has the same computational complexity as EF-RLS while guaranteeing that, under zero excitation, the difference between the covariance matrix
$P_{k}$
and
$P_{\infty }$
is asymptotically bounded. A numerical example shows that ER-RLS and CR-RLS both perform nearly identically to EF-RLS under persistent excitation while protecting against covariance windup when persistent excitation is lost.